Easy Mathematics for Biologists

Easy Mathematics for Biologists Easy Mathematics for Biologists Peter C.Foster Department of Applied Biology Univers...

Author: Peter C. Foster

109 downloads 1859 Views 1MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

Easy Mathematics for Biologists

Easy Mathematics for Biologists

Peter C.Foster Department of Applied Biology University of Central Lancashire Preston UK

harwood academic publishers Australia • Canada • France • Germany • India • Japan Luxembourg • Malaysia • The Netherlands • Russia Singapore • Switzerland

Copyright © 1998 OPA (Overseas Publishers Association) N.V. Published by license under the Harwood Academic Publishers imprint, part of The Gordon and Breach Publishing Group. All rights reserved. First published 1998 This edition published in the Taylor & Francis e-Library, 2003. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system, without permission in writing from the publisher. Printed in Singapore. Amsteldijk 166 1st Floor 1079 LH Amsterdam The Netherlands

British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. ISBN 0-203-30430-6 Master e-book ISBN

ISBN 0-203-34408-1 (Adobe eReader Format) ISBN: 90–5702–339–3 (softcover)

Contents Preface

1

2

Introduction

1

1.1 1.2

1 1

4

6

Measurements and Units Fractions Decimals Percentages Problems

3 3 4 6 8 10

Ratio and Proportion: Amounts, Volumes and Concentrations

13

3.1 3.2 3.3 3.4 3.5 3.6

13 16 18 20 23 25

Introduction to Ratios and Proportions Molarity Dilutions Concentration, Volume and Amount Concentrations and Dilutions in Practice Problems

Exponents and Prefixes: Scientific Notation, Conversion of Units and More on Concentrations 29 4.1 4.2 4.3 4.4

5

How to Use this Book Understanding a Problem—What You Need to Know

Measurements and Units: Fractions, Decimals and Percentages 2.1 2.2 2.3 2.4 2.5

3

vii

Introduction Exponents and Scientific Notation Metric Prefixes: Changing the Units Problems

29 29 33 35

Solving Equations and Evaluating Expressions

39

5.1 5.2 5.3

39 43 45

Solving Equations Evaluating Expressions Problems

Logarithms

49

6.1

49

Logarithms and Exponents v

vi

Contents

6.2 6.3 7

8

Manipulating Logs Problems

51 56

Straight-Line Graphs: Calibration Curves and Linear Rates of Change

59

7.1 7.2 7.3 7.4

Proportional Relationships Equations of Straight Lines Equations of Straight Lines and Straight-Line Graphs in Practice Problems

59 62 66 69

Non-Linear Rates of Change: Graphs, Transformations and Rates

73

8.1 8.2 8.3

73 79 84

Graphs that are not Straight Lines, and their Transformation Rates of Change Problems

Answers to Problems Index

87 105

Preface

Over the past decade or so I have become increasingly concerned that many students on B.Sc. and HND courses in the biological sciences have great difficulty with numerical calculations relating to aspects of biology, whether it is with both the mathematics and the applications, or just a problem applying the mathematics to a particular situation. While many mathematics textbooks exist, these tend to be either too general and lacking in relevant examples, or too advanced, covering topics which are not required by most biology students. (I am using the term ‘biology’ to cover all the biologically-related disciplines such as ecology, zoology, botany, biochemistry, physiology, and microbiology.) This book is intended to be used primarily by such first year students to develop their skills in this area. Rather than provide a conventional textbook, I have chosen to write this as a self-contained workbook. The aim is for most students to be able to work through the book independently of any mathematics lectures. Alternatively, it could be used as part of a variety of courses, whether skills-based or subject-based. In my own university I have used it in place of lectures as the basis of a numeracy skills module for a variety of biology students. The students also received some tutorial help. The book is arranged in chapters which lead from the basic mathematical ideas of fractions, decimals and percentages, through ratio and proportion, multipliers, exponents and logarithms, to straight line graphs, and finally to graphs that are not straight lines, and their transformation. The associated applications covered are the types of problem most commonly encountered in degree and HND courses in the biological sciences and include concentrations and dilutions, changing units, pH, and linear and non-linear rates of change. Each chapter starts with an explanation of the mathematical concepts covered together with worked examples. The main concepts or definitions to be remembered are highlighted. Worked examples of the applications in biology follow the ‘pure’ mathematics sections. The applied examples obviously require some understanding of the science as well as the mathematics; I have tried to explain the science to the extent necessary to tackle the problems. At the end of each chapter are numerous examples of both pure and applied problems. Answers to the problems are at the end of the book. I am indebted to a number of people who have made helpful suggestions and encouraged the development of the book. Peter Robinson, Philip Roberts and Sally Foster read drafts of the text and made useful suggestions. Many of the students who used a draft version also either provided helpful comments or showed me what they found most difficult. Sally Foster also pointed out ambiguities, suggested alternative ways of presenting some arguments, and identified typographical errors and errors of grammar and punctuation. Peter Robinson helped with any information technology problems I had. vii

1 INTRODUCTION

1.1

HOW TO USE THIS BOOK

You are probably using this book because you have found you have some difficulty with numerical calculations relating to aspects of biology. You are not alone! Many students of the biological sciences have the same problem. Nevertheless it is important that you develop the ability to carry out various mathematical procedures and to understand quantitative information being produced as evidence to support some argument or hypothesis. The contents of this book represent the type of problems most commonly encountered in degree and HND courses in the biological sciences. Mastery of these will mean you should improve your confidence in understanding many parts of your course. The book is arranged in chapters which are sequential in that I believe you need to master the contents of one before you will be able to fully understand the next. Each chapter or part of a chapter starts with an explanation of the mathematical concepts covered, follows with examples that are purely mathematical, and ends with examples that are applications of the mathematics in biology. You may feel competent at doing the calculations in the first chapter(s) and be tempted to ignore them. You may have difficulties with both the mathematics and the applications, or just have a problem applying the mathematics to a particular situation. Whatever the case, I suggest you read each chapter and try out the ‘pure’ mathematics examples marked‘*’. If you get these correct, try all of the applied problems. If you also get these correct, move on to the next chapter. If you do not get the right answers, reread the explanations and try the rest of the ‘pure’ problems, and then repeat your attempts at the applied problems. Answers to the problems are at the end of the book. Try to resist the temptation to look at the answers before committing yourself to an answer in writing. It is very easy to fool yourself that you can do a problem if you look at the answer first!

1.2

UNDERSTANDING A PROBLEM—WHAT YOU NEED TO KNOW

It is quite likely that you are able to do many of the ‘pure’ mathematical problems in the following chapters, but that your problem is working out how to manipulate 1

2

Easy Mathematics for Biologists

the data you are presented with in a real situation. The following example of a typical situation in biochemistry may help to illustrate how to understand the problem and dissect out the relevant parts. Suppose you are conducting an experiment to measure the rate of an enzymecatalysed reaction. (An enzyme speeds up a specific chemical reaction.) You want to find out how the rate of reaction changes if you vary the concentration of one of the reactants, called PEP. You know you will want to measure the rate at about 9 different concentrations, and that the concentrations have to start at about 0.2 mmol/ 1 and go down by a factor of about 100. The final volume of the solution in which you are measuring the rate is 2.0 ml. What volumes and concentrations of PEP are you going to use? What you need to understand is:

• • • •

what is meant by concentration, by volume, and by amount, and their interrelationships; what 0.2 mmol/1 means; how to convert multiples of one unit to another; that there is a physical limit to the volume of a solution you can pipette, determined by your equipment.

What you do not need to know to do this problem is anything about the enzyme or reaction, so don’t worry about these. The first three above require that you understand the mathematical ideas of fractions, decimals, proportions, exponents and units. These are covered in the first few chapters and, together with straight line graphs, are the most important concepts you need to understand and be able to use. The later chapters cover other topics which are also important, but less basic.

2 MEASUREMENTS AND UNITS: FRACTIONS, DECIMALS, AND PERCENTAGES

2.1

MEASUREMENTS AND UNITS

The quantities you will come across most often are length, mass, amount of substance, time and temperature. Measurements of these consist of a number and a unit. The number expresses the ratio of the measured quantity to a fixed standard, and the unit is the name of the standard. For example, if a piece of wood is 3 metres long it is 3 times the length of the standard used to define 1 metre. It is vitally important that you use units correctly and especially that you write down the correct unit after calculating a numerical answer. (Some measurements do not have units. This may be because they are ratios of two measurements with the same units, or because they are a particular kind of mathematical transformation of a measurement called a logarithm. Logarithms are explained in Chapter 6.) A large number of different units could be used. Distance could be measured in metres, inches, hands, feet, furlongs, miles, or light-years! The accepted convention for use in science is the ‘SI’ system in which there are ‘base’ units and ‘derived’ units each of which has a specified abbreviation or symbol. The base units for length, mass, amount of substance, time and temperature are metre (m), kilogram (kg), mole (mol), second (s) and kelvin (K) respectively. Derived units are made up from combinations of base units; for example, the unit of energy, the joule, is metres squared times kilograms divided by seconds squared. It is very important that you write down the appropriate unit with the number and that you use the correct abbreviation or symbol. Otherwise, you may confuse the reader or the information you are giving may be meaningless. Measurements should be written with the number separated from the unit by a space, and the unit should be singular and without the full stops normally used to show an abbreviation. For example, a mass of 6 kilograms is written as 6 kg and not as 6 kg or 6 kg. or 6 kgs. Where units are combined there is a space between them. For example, a speed of 6 metres per second is written as 6 m s-1, and not as 6 ms-1 or 6 m.s-1. (s-1 is shorthand for ‘per second’: 6 m s-1 is the same as writing 6 m/s.) An area of 4 square metres is written as 4 m2 and not as 4 m2 and a volume of 2 cubic metres is written as 2 m3. (Note: m2 is shorthand for m×m, square metres,

3

4

Easy Mathematics for Biologists

the area of a square of side 1 metre. Similarly, m3 is shorthand for m×m×m, metres cubed, the volume of a cube of side 1 metre.) Although using SI makes calculations easier in some ways, it is not perfect, and most biologists are sensible enough to use other units if there is a good reason. For example, when reporting changes over days or years it would be silly to use seconds as the unit. Particularly common is the use of the litre as a measure of volume for the reasons that it is a convenient size; laboratory glassware is calibrated using it; in most countries (and increasingly in the UK) people are familiar with it; and it is readily convertible to the SI units. The litre is normally abbreviated as ‘l’, but many American texts use ‘L’. There are one thousand litres in a cubic metre. (A litre is the volume of a cube whose sides are one tenth of a metre by one tenth of a metre. One tenth of a metre is also known as a decimetre (dm), so a litre is also the same as a cubic decimetre.) We will come back to more about units in later sections, but now I want to move on to ways of expressing quantities that are not whole numbers.

2.2

FRACTIONS

A fraction is a whole number divided by another whole number other than 0, e.g. . (Note that the symbol ‘/’ means ‘divided by’ whether used in a simple fraction or a more complex situation.) The number above the line (3) is called the numerator and that below the line (5) is called the denominator. Two fractions are equal if one can be converted to the other by multiplying or dividing both the numerator and the denominator by the same number. These are called equivalent fractions. Example 2.1

Example 2.2

To add fractions you need to get the denominators the same. Example 2.3

Measurements and Units

5

Example 2.4

Example 2.5

Similarly, to subtract fractions, the denominators must be the same. Example 2.6

When multiplying fractions, multiply the numerators; then multiply the denominators; then if possible reduce by dividing both the numerator and denominator by any whole number that will leave a whole number. You may also be able to cancel numbers appearing as factors of both the numerator and denominator. Example 2.7

Example 2.8 Note that in this example you could have reduced earlier:

6

Easy Mathematics for Biologists

Example 2.9

Cancelling at the beginning is particularly useful when you are multiplying or dividing several fractions or a complex fraction: Example 2.10

The way that you cancel does not matter so long as whatever you do to the numerator, you do to the denominator. To divide fractions, multiply the first fraction by the reciprocal (i.e. by the fraction turned upside down) of the second and then reduce or cancel wherever possible. Example 2.11 Example 2.12

Example 2.13

Example 2.14

2.3 DECIMALS

Although many calculations involve fractions, the answer is usually presented in a decimal form rather than as a fraction, e.g. as 0.375 g, not 3/8 g. Remember that in decimals, each position going further to the right of the decimal point is one tenth of the previous one, and each to the left is 10 times bigger.

7

Measurements and Units

(hundreds) (tens) (units) . (tenths) (hundredths) (thousandths) so 475.356 could be written as the fractions 475

or as

When multiplying decimals you can first ignore the decimal points and multiply the numbers. Then count the number of digits to the right of the decimal points in the original numbers. This will be the number of digits to the right of the decimal point in the answer. Example 2.15

To divide a decimal by a decimal, keep multiplying both numbers by 10 until the divisor (the number you are dividing by) is a whole number. Then divide as for a whole number. The decimal point is above the decimal point in the number being divided. Example 2.16

In practice, you will almost certainly prefer to use a calculator for these kinds of calculations. Whether you do or not, you should get into the habit of making an estimate of the answer as a check on your calculation. In the above example, if you were to use a calculator, you should first estimate the answer by rounding the is between and . is 17.5 and is about 23.3, so the numbers: answer should be between these, i.e. about 20.

8

Easy Mathematics for Biologists

2.4 PERCENTAGES

Percentages are a particular kind of fraction where the denominator is 100 but is indicated by the % sign. Example 2.17 To convert any fraction or decimal into a percentage, multiply by 100. Example 2.18 Example 2.19 To convert a percentage to a fraction or a decimal, divide by 100. Example 2.20 To calculate a percentage, first calculate the fraction, then multiply by 100. Example 2.21 In a group of 42 students, 10 are male. What percentage are male?

To apply a percentage to a number, multiply by the number % and divide by 100. Example 2.22

If 51% of the population of the UK is female, and the total population is 56 million, what is the female population?

You will often come across a percentage increase or decrease. Example 2.23

The number of unemployed fell from 2 312 000 to 2 112 000. What percentage fall is this?

Example 2.24

The number of unemployed rose by 1.7% last month from 2 312 000. What is the new figure?

Measurements and Units

9

An alternative and quicker way of doing this is to realise that the original number is 100%, so the new number will be (100%+the percentage increase)×the original number.

Example 2.25

If the number falls by 1.7% of this new figure next month, what will the number be then?

Note that this answer is not the same as the original number. A 1.7% increase followed by a 1.7% decrease does not cancel out because the number you are taking the percentage of has changed. It is important in problems that you are clear to what number a percentage refers. One way of describing the concentration of a chemical solution is to say the weight of chemical in a certain volume of solution. For example, the appropriate concentration for a pesticide might be achieved by dissolving a sachet containing 50 g in a volume of 1 gallon of water. If the final volume is also 1 gallon, the concentration is therefore 50 g per gallon. In biology, concentrations of solutions are often described in terms of weight per volume, usually when the molecular weight of the solute is not known, e.g. protein. It is usual to quote the concentration as a number of g, (x g), per 100 ml and this is abbreviated to x%(w/v). The (w/v) stands for weight/volume, indicating it is the weight (g) divided by the volume (100 ml). (Note that 1 ml is 1 millilitre which is one thousandth of a litre.) Example 2.26

How would you prepare 250 ml of a 4%(w/v) solution of NaCl in water?

Example 2.27

If 5 g of NaCl is dissolved, then diluted with water to make 250 ml, what is the %(w/v) concentration of NaCl in the solution?

10

Easy Mathematics for Biologists

It is also possible to describe concentrations as x%(v/v) or x%(w/w). For example, a 50% (v/v) ethanol solution is made by mixing 50 ml ethanol with 50 ml water. A 5% (w/w) sodium chloride solution is made by mixing 5g sodium chloride and 95g water. Note that this is not quite the same thing as 5%(w/v) which is 5g sodium chloride in 100 ml solution final volume, so you would need to dissolve the salt and make up to volume in a volumetric flask. Example 2.28

How would you prepare 15 ml of a 5%(v/v) solution of ethanol in water? You need 15 ml×5 ml/100 ml =75 ml/100=0.75 ml of ethanol So add 0.75 ml of ethanol to 14.25 ml of water.

Example 2.29

What is the %(v/v) concentration of a solution of ethanol made by mixing 25 ml ethanol with 225 ml of water?

2.5 PROBLEMS

If you have just finished reading this chapter I suggest you now try the ‘pure mathematics’ examples marked‘*’. If you get these correct, try all of the applied examples 31–36. If you also get these correct, move on to the next chapter. If you do not get the right answers, re-read the explanations and try the rest of the problems 1–30, and then repeat your attempts at the applied problems. Do each of the following calculations without, then check with, a calculator. 1.

2.

3.*

4.

5.

6.

7.

8.

9.*

10.

11. 17.4+13.26

12. 17.4-13.26

13. 15.01–13.97 16. 9.84÷8

14. 7.7×3.5 17. 13.1÷3.8

15.* 9.8×17.31 18. 177.68÷13.5

Measurements and Units

11

You can use a calculator for the following problems, but make sure you estimate the answers first. 19. 18.60÷3.41 22. 15% of 46 25. 0.75=?%

20. 18.60÷0.783 23. 11.2% of 31.3 26. =?%

21.* 3% of 21 24.* 15 is ?% of 45

27. 81 increased by 9% gives what? 28.* What when increased by 20% gives 72? 29. 35 decreased by 20% gives? 30.* What, when decreased by 75%, gives 20? 31. The dry weight of the seeds of an annual weed is 1.75 g and of the rest of the plant 5.21g. What percentage of the total dry weight are the seeds? 32. A typical bacterial cell is 70% water, 15% protein, and 7% nucleic acid. What weight of protein and nucleic acid is there in 3g of bacteria? What will be the weight of matter that is not protein, nucleic acid, nor water? 33. Complete Table 2.1 which shows the relative proportions or amounts of some elements in the human body. 34. What percentage of the weight of disodium hydrogen phosphate, Na2HPO4, is from phosphorus? (Use the following atomic weights of the elements: Na=23, H=1, P=31, O=16) Also express this percentage as a fraction and as a decimal.

Table 2.1 Data for Question 33, Chapter 2.

12

Easy Mathematics for Biologists

35. How would you make up the following sucrose or ethanol solutions? (i) 100 ml of a 3% (w/v) solution of sucrose, (ii) 500 ml of a 3% (w/v) solution of sucrose, (iii) 500 ml of a 5% (w/w) solution of sucrose, (iv) 500 ml of a 0.5% (v/v) solution of ethanol. (v) 20 ml of a 10% (w/v) solution of ethanol. 36. For each of the first three solutions in 35 above, calculate the weight of sucrose there would be in 5 ml.

3

3.1

RATIO AND PROPORTION: AMOUNTS, VOLUMES AND CONCENTRATIONS

INTRODUCTION TO RATIOS AND PROPORTIONS

A ratio is one number divided by another number but it can be written in several ways:

Example 3.1

If there are 20 staff and 400 students, the student: staff ratio=400:20

Problems involving proportions (in a purely mathematical sense) can be represented by equations involving two ratios. Example 3.2

If on a map, 1 cm represents 2 km, how many cm represent 12km? The scale of the map is a ratio of the distance on the map to the distance on the ground, and this ratio must be the same for any distance, so the problem can be represented as an equation of two ratios in which ‘x’ stands for the unknown number:

(You may find it helpful to read the first part of section 5.1 if you are unsure about solving simple equations.) Since equations stay the same if you do the same thing to each side, to find the value of x, multiply each side by 12 km:

13

14

Easy Mathematics for Biologists

In the example above, you could have multiplied by both denominators: 1 cm×12 km=xcm×2 km From this you can see that if the quantities in an equation of two ratios are represented by a, b, c and d:

Example 3.3

Example 3.4

Example 3.5 In a pharmacology experiment a piece of guinea-pig ileum is attached to a transducer so that a small contraction of the tissue produces a larger movement of a pen on a chart recorder. If a 3 mm movement of the transducer produces a 13 mm deflection on the chart recorder, what size of contraction is responsible for a 33 mm deflection on the chart recorder? The two ratios of tissue contraction to pen movement must be the same, so if x stands for the required value,

The most common calculations involving ratios and proportions are to do with unit conversions, and with concentrations and dilutions. Example 3.6 1 inch = 2.54 cm How many cm in 18 inches?

Ratio and Proportion

15

Example 3.7 100 ml contains 5g of glucose. How much is there in 20 ml?

Example 3.8 To 5.0 ml of a 5% solution of glucose is added water to a final volume of 250 ml. What concentration is this new solution? You can probably do this in your head. 5.0 ml becomes 250 ml which is a 50 times bigger volume, so the concentration must be

Working it as a proportion though you might be tempted to write:

This is wrong. The volume is not proportional to the concentration, but inversely proportional, i.e. the larger the volume, the lower the concentration. Therefore the relationship is:

I think it is easier to write the relationship in the form in which you would say it: 5 ml of 5%=250 ml of x% For ‘of’ you can substitute ‘times’ (just as ‘per’ means ‘divided by’).

16

Easy Mathematics for Biologists

A better way of writing out the relationship would be to include the units:

A good reason for writing out the problem in this way is because the units might be different. For example, suppose that to 5 ml of a 5% solution is added water to a final volume of 5 litres. What concentration is the new solution?

Notice that the units do not balance, so you need to convert litres to ml. Since there are 1000 ml in 1 litre, there are 5000 ml in 5 litres.

3.2 MOLARITY

So far I have expressed chemical concentrations only in terms of weight per volume, weight per weight, or volume per volume. A more common and usually more useful way is in terms of molarity, i.e. the moles per volume. A mole (symbol mol) is the SI unit of amount of substance. For an element the mass of a mole of that element is its atomic mass expressed in grams instead of atomic mass units. For example, the atomic mass of carbon is 12.011, so a mole of carbon has a mass of 12.011g. It follows that a mole of any element contains the same number of atoms as a mole of any other element. This number, which is called Avogadro’s number, is 602 000 000 000 000 000 000 000. Similarly, for a compound, a mole is the relative molecular mass (molecular weight or formula weight) in grams. For example, 1 mole of glucose (C6H12O6) contains Avogadro’s number of molecules and has a mass of 180g because its formula weight is 180. (Formula weight=(6× 12)+(12×1)+ (6×16)=180.)

Ratio and Proportion

17

Example 3.9

Calculate the mass of 0.25 moles of Na 2HPO 4. (Use the following atomic weights of the elements: Na=23, H=1, P=31, O=16) Relative molecular mass (RMM) or formula weight= (2×23)+(1×1)+(1×31)+(4×16)=142g/mole so 0.25 mol×142g/mol=35.5g.

Example 3.10

Calculate the number of moles of NaOH in 15g. Relative molecular mass =23+16+1=40g/mol So in 15g there are moles.

If a solution contains 180g glucose in 1 litre then it has a concentration of 1 mol/1. A solution containing 360g sucrose in 1 litre also has a concentration of 1 mol/1 because the relative molecular mass of sucrose is 360. The two solutions therefore have the same number of molecules per litre, but the glucose solution is 180g/1 which is the same as 18g/100 ml or 18% (w/v) whereas the sucrose solution is 36% (w/v). In general, then, two solutions of different compounds but the same molar concentration (molarity) have the same number of molecules per litre, but usually different masses of solute per litre. A solution containing 360g of glucose in 1 litre would have 2 moles/litre and therefore be 2 molar. Note that the unit mol/1 can also be written mol 1-1 or M, so a 0.2 mol/1 solution is the same as a 0.2 mol 1-1 solution and a 0.2M solution, the M being read as molar. Example 3.11 What is the molarity of a solution made by dissolving 8g of NaOH in water and diluting to 2 litres?

Example 3.12 What is the molar concentration of a 0.9%(w/v) NaCl solution?

Example 3.13 How many moles of a compound are present in 250 ml of a 0.2 mol/1 solution?

18

Easy Mathematics for Biologists

Number of moles=volume×concentration =250 ml×0.2 mol/1000 ml=50/1000=0.05 moles Example 3.14 What weight of glucose is needed to make 1 litre of a 0.25 M solution? 0.25 M=0.25 mol/1 so you need 0.25 mol of glucose for 1 litre. Since glucose has a molecular weight of 180g/mol, weight of glucose needed =0.25 mol×180g/mol=45g Example 3.15 What weight of glucose is needed to make 250 ml of a 0.5 M solution? 0.5 M=0.5 mol/l so you need 0.5 mol for 1 litre 0.5 mol×180g/mol=90 g for 1 litre. But you only want 250 ml, therefore you need 90g×250 ml/1000 ml=22.5g. Rather than writing all this out, it could be combined into one expression:

Look carefully at the units. Notice that they all cancel out except for g. If you write the units in at each step of the calculation then cancel them, you can check that the calculation is correct. This is one way to make sure you do not divide by something when you should be multiplying by it. Also it is important that you report your answer with the appropriate units. If the units are wrong, the calculation is wrong.

3.3 DILUTIONS

It is often more practical or more convenient to make up a ‘stock solution’ and then dilute that to the required concentration, than to make up the required solution directly. For example, you might want to make up 1 ml of a solution of a drug (in water) at a concentration of 1 millionth of a gram per ml, but the smallest amount you can weigh accurately is 1 hundredth of a gram. It would be a bit silly to weigh out this amount and make the solution up to 10 litres (10 000 ml) to make the required concentration.

Ratio and Proportion

19

A more sensible approach is to weigh out g and make the solution up to 100 ml. The concentration of this ‘stock solution’ is

This is 100 times more concentrated than the solution needed. You now need to dilute a portion of this solution by a factor of 100, i.e. what is called a 100-fold dilution or 100×dilution. A practical way of doing this would be to add 0.1 ml of the stock solution to 9.9 ml of water.

so the final solution is

of the concentration of the stock solution.

This method uses only a single dilution step, but often it is necessary or convenient to use two or more dilution steps. This is known as ‘serial dilution’. Example 3.16 Prepare 10 ml of a 0.0001 mol/1 solution (in water) of glucose from a 1 mol/l stock solution. You can do this easily by serial dilution. Take 0.1 ml of stock+9.9 ml of water. This is a 100-fold dilution, so the concentration of this solution (A) is 0.01 mol/1. Now take 0.1 ml of (A) and add 9.9ml of water. Again, this is a 100-fold dilution, so the concentration of this final solution is 0.01/100=0.0001 mol/1. Of course, there is no reason why dilutions should always be by factors of 10. The fold dilution is equal to the final volume divided by the volume of the original solution. Remember that the final volume is the volume of the original solution plus the volume of the additional diluent. Example 3.17 In a 4-fold dilution

Note that this means the volume of diluent you need is the final volume minus the volume of the original solution.

20

Easy Mathematics for Biologists

Example 3.18 To make 50 ml of a 25-fold dilution, what volumes should be used?

Therefore add 2 ml of original solution to 48 ml diluent to get 50 ml of a 25-fold dilution. Example 3.19 How much has a solution been diluted if 3 ml has 15 ml diluent added?

There are other ways of expressing dilutions. Rather than saying ‘a 10-fold dilution’ for example, you could say ‘a 1 in 10 dilution’ or ‘a 1+9 dilution’. The term ‘dilution factor’ is usually used to mean the same as ‘fold dilution’.

3.4 CONCENTRATION, VOLUME AND AMOUNT

Calculations for making up solutions, making dilutions and measuring the amount of substance produced all involve a simple relationship between concentration, volume and amount.

or, amount=concentration×volume e.g. 0.5 mol=0.5 molar×1 litre

or,

Ratio and Proportion

21

It is really important that you distinguish between amount and concentration. For example, if you start with 1 mole of glucose and dissolve it in water to make 1 litre, the concentration is 1 mol/1 (or 1M), the amount is 1 mol, and the volume is 1 litre. If you then remove 500 ml, the volume is 500 ml, the concentration is still 1 mol/1, but the amount present is 0.5 mol. If you then add 500 ml of water, the volume becomes 1 litre, the amount stays at 0.5 mol, but the concentration falls to 0.5 mol/1. (See Fig. 3.1) Example 3.20 It was found that the concentration of NADH in a 2.0 ml sample taken from a flask containing 250 ml was 0.001 mol/1. How much NADH was in the flask? NADH concentration=0.001 mol/1 Total volume at this concentration=250 ml=0.251 Therefore the total amount of NADH=0.001 mol/1×0.251=0.00025 mol. Example 3.21 A 10 ml sample was removed from a flask containing 500 ml of an aqueous solution of X. To 1.0 ml of this sample was added 4.0 ml water, and then 2.0 ml of this final solution was put in a cuvette to measure the concentration of X. This was found to be 0.03 mol/1.

FIGURE 3.1

Relationship Between Amount, Volume and Concentration.

22

Easy Mathematics for Biologists

How much X was in the flask? The concentration of X in the cuvette=0.03 mol/1. This has been diluted 5 fold, so the original concentration of X=0.03 mol/1×5= 0.15 mol/1. Total amount of X in the flask =0.15 mol/1×500 ml =0.15 mol/1×0.51 =0.075 mol. Example 3.22 Using the previous example, complete Table 3.1 showing the volume, amount and concentration of each of the solutions. You can find the missing information by working down the table. Table 3.1 Data for Example 3.22.

Ratio and Proportion

3.5

23

CONCENTRATIONS AND DILUTIONS IN PRACTICE

Often you will be interested in measuring the effects of different concentrations of a substance on some system, so you need to work out how to arrange this. Suppose the system you are using (e.g. a chemical reaction) requires that you always have a 2.0 ml volume in the tube in which the system occurs. This means that you need to work out how to achieve a range of concentrations of your substance in the 2.0 ml volume, bearing in mind first that if you start with 2.0 ml and add a volume of your test substance the system volume will increase, and second, that adding a volume of your test substance to a volume of another solution will reduce the concentration of the test substance. Look back at the example in Section 1.2 where you have a system volume of 2.0 ml. The highest concentration of PEP you want to use is 0.0002 mol/l and you need 9 different concentrations of PEP covering a 100-fold range. The first thing to do is to think about the volumes of solution it is practical to pipette accurately. You can assume that you could use pipettes to cover a range of volumes from 10.0 ml to 0.01 ml. There are two main ways of proceeding.

(1) Preparing a Series of Dilutions (a) Decide what is a convenient volume to pipette, e.g. 0.1 ml, and make up a series of different concentrations of PEP so that you always add a fixed volume (0.1 ml) of various concentrations to the system. You need a 1.9 ml volume of the other components of the system so that when you add 0.1 ml, you get a final volume of 2.0 ml. Since you are adding 0.1 ml of PEP and ending up with 2.0 ml, the concentration of PEP is being reduced by a factor of 20. Therefore the concentration of PEP that you make up must be 20×the concentration that you want to achieve in the system. Therefore the highest concentration of PEP that you make up must be 0.0002 mol/1× 20=0.004 mol/1 and the lowest must be a 100fold dilution of this. You could make up a stock solution of PEP that is 0.004 mol/ 1, and from it make your series of dilutions. Example 3.23 From a stock solution of 0.004 mol/1, a series of 9 concentrations can be made as in Table 3.2. (b) Alternatively you could make a series of different PEP concentrations by the method of ‘doubling dilutions’. Add 0.5 ml water to each tube except the first, which contains the 0.004 mol/1 stock solution of PEP. Remove 0.5 ml from this tube and pipette into the second, and mix. As there is now 1.0 ml of solution in this second tube, the concentration will have been halved, being now 0.002 mol/1.

24

Easy Mathematics for Biologists

Table 3.2 Preparing a Series of Dilutions.

Pipette 0.5 ml from this tube into the third tube. This again halves the concentration so producing a 0.001 mol/1 solution. Continue in this way for the whole set of tubes. You will end up with a series of tubes with each concentration being one half of the previous one: 0.004, 0.002, 0.001, 0.0005, 0.00025, 0.000125, 0.0000625, 0.00003125, and 0.000015625 mol/1. Notice that it only requires 8 tubes to cover the 100-fold range, but the numbers would be rather awkward to handle if you were going to plot a graph of the results.

(2) Using Different Volumes of a Stock Solution A different approach is to pipette different volumes of your stock solution, plus different volumes of water, directly into the system. This time the concentration of the stock solution you make up needs to be lower because you will start by pipetting a larger volume, and the volume of the other components of the system must be smaller. Example 3.24 Using a stock solution of PEP that is 0.004 mol/1, the required system concentrations can be made by adding stock and water to 1 ml of the other components as in Table 3.3.

Ratio and Proportion

25

Table 3.3 Using Different Volumes of a Stock Solution to Prepare Dilutions.

There are advantages and disadvantages with each method. Using the method (1a) involves three pipettings for each concentration, and the use of more tubes. Method (2) involves only two pipettings (so should be more accurate) and no tubes, but if you end up pipetting very small volumes accuracy may suffer. The method of doubling dilutions is attractive because you do not need to think about changing the volumes you are pipetting, but it involves three pipettings for each concentration, and any error in one tube will be carried on to affect subsequent ones. In microbiology or other cell culture experiments, serial dilution by a factor of 10 is often used to dilute a cell culture enough so that the number of cells per ml can be counted, or a dilute culture spread on a plate so that each cell grows to form a separate colony. A typical procedure would be to set up a series of tubes containing 9 ml of medium, and to add 1 ml of the original culture to the first, mix, and pipette 1 ml of this into the next tube, and so on. The number of cells per ml in each tube will then be of that in the previous tube. In this way it is possible to prepare quickly a wide range of concentrations.

3.6. PROBLEMS

If you have just finished reading this Chapter I suggest you now try the ‘pure mathematics’ examples marked‘*’. If you get these correct, try all of the applied examples. If you also get these correct, move on to the next chapter. If you do

26

Easy Mathematics for Biologists

not get the right answers, re-read the explanations and try the rest of the problems 1–12, and then repeat your attempts at the applied problems. Write the following ratios as fractions in their simplest forms. 1. 12:6 4. 13:52 Solve for x. 7. x/9=4/3 10. 56/x=8/7

2. 24:15 5. 36g:84g

3.* 9:72 6.* 210 ml:71

8. 2/9=x/3 11. 1/x=0.5/1

9.* 11/8=44/x 12.* 5/x=3/0.5

13. If a map is drawn 1:25 000, what distance does 2 cm represent? 14.* What scale is a map where 1 cm represents 5 km? 15. How many litres in 1 gallon if 1 pint=567.5 ml and there are 8 pints in 1 gallon? 16. Ecologists sometimes use the capture/recapture technique to estimate the population of an animal in a certain study area. For example, for voles the method consists of trapping a sample of voles in traps scattered throughout the area. The trapped voles are marked in a way which enables them to be identified but does not affect them. The marked voles are released and the traps reset a day later. The total number of voles captured this time, and the number recaptured which are marked, can be used to estimate the total population. The proportion of marked voles recaptured to total marked voles is the same as the proportion of total voles captured to the total population. (Note that this assumes there is no change in the habits, life-expectancy, etc. of the marked voles.) If initially 50 voles are captured, marked, and released, and later 40 voles are captured, of which 5 carry the mark, what is the total vole population in the area surveyed? 17. Given a 5% (w/v) solution of sodium chloride (formula weight= 58.5g) complete Table 3.4. 18. The formula weight of ATP (disodium salt) is 551. (i) What is the weight of 0.5 mol of this? (ii) What weight of this ATP is required for 1 litre of a 0.5 mol/l solution? (iii) What weight of this ATP is required for 200 ml of a 0.25 mol/l solution? (iv) What is the % (w/v) concentration of a 0.5 mol/l solution of ATP? (v) What is the % (w/v) concentration of a solution of ATP containing 0.0187g in 50 ml?

Ratio and Proportion

27

Table 3.4 Data for Question 17, Chapter 3.

(vi)

19.

What is the molar concentration of a solution of ATP containing 0.0374g in 30 ml? (vii) What volume of 0.01 mol/1 ATP solution could you make with 1g of ATP? (viii) What volume of 0.005 mol/1 ATP solution could you make from 0.2 ml of a 1% (w/v) solution? (ix) Given 100 ml of 0.01 mol/1 ATP solution, how would you make up a set of solutions of 10 ml volume each, with concentrations of 0.005, 0.004, 0.003, 0.002, 0.001, and 0 mol/1? (x) If 0.05 ml of each of the six solutions in (ix) above were pipetted into a cuvette containing 1.95 ml of other solutions, what would be the final ATP concentration in each cuvette? Given 100 ml of a 0.5 mol/1 solution of glucose (formula weight 180), how would you prepare the following? (Assume the smallest volume you can measure accurately is 0.05 ml and that you need to be reasonably economical with the stock solution.) (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)

20.

10 ml of a 0.02 mol/1 solution. 10 ml of a 0.004 mol/1 solution. 1.0 ml of a 0.01 mol/1 solution. 1.0 ml of a 0.03 mol/1 solution. 10 ml of a 0.1% (w/v) solution. 5.0 ml of a 0.03% (w/v) solution. 2.0 ml of a 0.02% (w/v) solution. 1.0 ml of a 0.15% (w/v) solution. A set of 5 tubes containing 1.0 ml of glucose solutions with the concentrations 0.02, 0.015, 0.01, 0.005, and 0 mol/1. (x) A set of 5 tubes containing 10 ml of glucose solutions with the concentrations 0.20, 0.15, 0.10, 0.05, and 0% (w/v). Inulin (not to be confused with insulin) is a soluble polysaccharide that if injected into a vein, is filtered from the blood plasma by the glomeruli of

28

Easy Mathematics for Biologists

the kidneys in the same concentration as in plasma. It is neither reabsorbed nor excreted in the tubules, and so appears in the urine. Therefore its rate of disappearance from the plasma is a measure of the ‘glomerular filtration rate’ (GFR, the volume filtered by the glomeruli per minute) as is its rate of appearance in the urine. Measurement of GFR can give a useful indication of kidney function. Calculate the GFR in ml/minute if the rate of inulin excretion into the urine is 120mg/minute when the plasma concentration is consta nt at .

4 EXPONENTS AND PREFIXES: SCIENTIFIC NOTATION, CONVERSION OF UNITS, AND MORE ON CONCENTRATIONS

4.1 INTRODUCTION

You may have noticed that I have so far avoided using many multiples of units. For example, for volumes I have only used litres or millilitres, and for weights, grams or kilograms. This was because I assumed you would be familiar with these units and could confidently convert from one to the other, but may not be so competent at using others. However, you may also have noticed that in restricting myself I have had to write some cumbersome numbers, e.g. 2 312 000, 10 000 ml, and 0.0001 mol/l. To avoid writing numbers like this I could have expressed them either in scientific notation or by using a different unit. To use either you need to understand exponents.

4.2 EXPONENTS AND SCIENTIFIC NOTATION

You can express many numbers as being the same as other numbers multiplied together, i.e. as the product of factors. Example 4.1

Example 4.2

Shorthand ways of writing these examples are:

Notice that in each case, the shorthand version consists of the number that is being multiplied by itself, which is known as the base, and a raised number representing how many times the base is written down, which is called the exponent 29

30

Easy Mathematics for Biologists

or power. (Powers of 2 are usually read as, for example, ‘10 squared’ and powers of 3 as ‘10 cubed’. 24 can be read as ‘2 to the 4th power’ or ‘2 to the power 4’ or more simply ‘2 to the 4’.) Example 4.3 For the number 24, the base is 2 and the exponent or power is 4. For any non-zero number a and any non-zero whole number b, ab=a×a×a×a…. where a is written b times. To multiply exponential numbers having the same base, add the exponents. For any non-zero base a and any non-zero whole numbers b and c, ab×ac=ab+c Example 4.4

Example 4.5

Note that to do this, the bases must be the same. If they are not, it may be possible to make them the same. Example 4.6

To divide exponential numbers with the same base, subtract the exponents.

Example 4.7

Example 4.8

What does 2-1 mean? For any non-zero base a and any non-zero whole number b,

Exponents and Prefixes

31

Example 4.9 What is 10-3 as a fraction?

Example 4.10 What is 3-4 ?

Example 4.11 Simplify 33/33 One number divided by itself obviously equals 1. Therefore any exponential number divided by itself is one.

In fact, for any non-zero base a, a0=1. Example 4.12 50=100=20=30=1 It is possible that you may need to simplify a number that is a power of an exponential number. Example 4.13

Example 4.14

If you write out values of the powers of 10 you can see that for each increase of 1 in the power, the value increases by a factor of 10. This is shown in Table 4.1. Table 4.1 Powers and Values.

Multiplying or dividing by 10b is therefore the same as moving the decimal point b places to the right or left respectively.

32

Easy Mathematics for Biologists

Example 4.15

3.0×103=3000 decimal point moved 3 places to the right 3.0×10-3=0.003 decimal point moved 3 places to the left 0.003×104=30 decimal point moved 4 places to the right

Example 4.16

0.003/10-4=? 10-4=1/104 Since dividing by a fraction is the same as multiplying by the inverse of the fraction, 0.003/10-4=0.003×104=30

Example 4.17

38.1=3810×10-2 or 381.0×10-1 or 3.81×101 or 0.381×102

Exponents become particularly useful when the base is 10 because it allows us to write out large numbers in a concise way. Example 4.18

Avogadro’s number=602 200 000 000 000 000 000 000. This can be written as 6.022×1023 because it is the same as 6.022×10×10×10×10 ...... (23 times).

In general, numbers written in the form a×10b where a is greater than 1 but less than 10 and b is a non-zero whole number are said to be expressed in scientific notation. To express a number in scientific notation, find ‘a’ by moving the decimal point to leave one non-zero digit to the left of the decimal point. Calculate the value of the exponent by counting the number of places you have moved the decimal point. If you have moved the decimal point to the left, b is positive, and if to the right, negative. Example 4.19

24 000=2.4×104 because the decimal point is moved 4 places to the left.

Example 4.20

0.000 024=2.4×10-5 because the decimal point is moved 5 places to the right.

To add or subtract numbers in scientific notation, you need to express them in the same power. Example 4.21

Exponents and Prefixes

33

To multiply or divide numbers in scientific notation, deal with the two parts separately. Example 4.22

Example 4.23

How you express the final value is a matter of choice. 1.68×10-1 is more concisely written as 0.168. Many people prefer to move the decimal point one or two places either way rather than write the exponential form. Example 4.24

320 rather than 3.2×102 32 rather than 3.2×101 0.32 rather than 3.2×10-1 0.032 rather than 3.2×10-2

If the value ends up having a power less than -2 or greater than +2, then you can either leave it in scientific notation, or change the units by using a prefix. This is covered in the next section. Before going on to this however, try the following example on your calculator. Example 4.25

3.6×103–0.8×103=?

The correct procedure is to key, in order: 3.6, EXP, 3, -, 0.8, EXP, 3, =. A common mistake is to key 3.6, ×, 10, EXP, 3, -, 0.8, ×, 10, EXP, 3 =, which gives an answer 10 times too big. EXP on the calculator stands for exponent with base 10, so you should not key in the×10. Similarly, if you wanted to divide by 104, you should key: ÷, 1, EXP, 4, and not ÷, 10, EXP, 4.

4.3 METRIC PREFIXES: CHANGING THE UNITS You have already come across the prefixes ‘milli’, meaning one thousandth and ‘kilo’ meaning one thousand. Prefixes like these are another way to avoid cumbersome numbers. Example 4.26

0.0004 m=4×10-4 m=0.4×10-3 m=0.4 mm

Example 4.27

400 000 g=4×105 g=400×103 g=400 kg

As you can see in the examples above, converting from one unit to another involves multiplying or dividing by a power of 10.

34

Easy Mathematics for Biologists

There are many prefixes. The most commonly used are shown in Table 4.2. Example 4.28

1 µg=1 microgram = 1×10-6 g or 0.000 001 g

It is unfortunate that some of the letters used as symbols for the prefixes are the same as, or similar to, those used for other prefixes or units. In particular, confusion can arise between G (giga) and g (gram), and between M, m, and µ (mega, milli, and micro) and m and M (metre and molar). It is therefore important that you use the correct symbols for the prefix and unit, do not invent your own abbreviations, and preferably use only correct SI units, except where others make more sense. It is usually best also to restrict yourself to multiples of 1000, e.g. kilo, milli, and micro. Below are some examples to show the problems. Example 4.29

‘a 10 mM solution…’

Since the M follows the m, it cannot mean mega, therefore it must mean molar, and so mM probably means millimolar since a ‘10 metremolar solution’ does not make much sense. It would be better to express this as ‘a 10 mmol/1 solution’ or ‘a 10 mmol 1-1 solution, or for strict SI enthusiasts, ‘a 10 mol m-3 solution’. Example 4.30

‘a rate of 5 µM 1-1m-1’

Table 4.2 Commonly Used Prefixes to Denote Multiples of Units.

Exponents and Prefixes

35

Since µM is micromolar, this appears to mean a rate of 5 µmol/1 per litre per metre, a concentration per volume per length, when what is probably meant is a rate of 5 µmol/l per minute, a change in concentration with time. Example 4.31

‘a concentration of 0.5 mmol dm-3…’

This is acceptable but could more simply be expressed as 0.5mmol/l, 0.5 mmol 1-1, or 0.5 mol m-3. Some people say that since a litre is not a proper SI unit, volumes should be based on a unit of length to the power 3. They then use dm3 or cm3 or mm3. However, this is not really in the spirit of SI either as one should not use powers of multiples of units. You do need to be able to convert between these units though. Example 4.32

Example 4.33

Example 4.34

What is the equivalent in litres of 1 mm3?

Concentrations in cells are typically millimolar or micromolar for metabolites of central pathways, and pico or femtomolar for messengers. Some equivalent units are:

4.4 PROBLEMS

If you have just finished reading this chapter I suggest you now try the ‘pure mathematics’ examples marked‘*’. If you get these correct, try all of the applied

36

Easy Mathematics for Biologists

examples. If you also get these correct, move on to the next chapter. If you do not get the right answers, re-read the explanations and try the rest of the problems 1– 59, and then repeat your attempts at the applied problems. Evaluate the expressions in questions 1–21. Use a calculator only to check your answers. 1. 4. 7. 10. 13. 16. 19.

23 (0.1)2 (2/3)2 a2×a3 a5/a3 (a2)3 103/102

2.* 5. 8.* 11. 14. 17.* 20.*

50 (0.3)4 (3/2)-2 a-2×a3 a-2/a3 (a2)-1 100×10-3

3. 6.* 9. 12.* 15. 18. 21.

3-3 (0.5)-1 (4/5)-4 a-3×a-2 a-3/a-2 (a-2)-3 102/1000

Express the following in scientific notation. 22. 0.000035 25. 373×10-6

23. 17 300 000 000 26. 0.016×104

24.* 3184.576 27.* 0.0351×10-2

Express the following as decimals. 28. 3.15×10-4 31. 1.5×10-2

29. 37.6×10-5 32.* 478.6×10–4

30.* 0.015×104 33. 0.0061×104

Evaluate the following (questions 34–45) using a calculator only to check the answers. 34. 36. 38. 40. 42. 44.

4.51×103+6.42×103 4.51×104–1.62×104 4.51×10–2-0.15×10-3 (4.5×103)(6.0×10-2) 3.6×10-5×11×102 3.6×10–3/12×10-2

35.* 37.* 39. 41.* 43. 45.*

1.32×108 +8.68×108 1.32×10–2-4.36×10-2 1.5×10–3-4.5×10-2 4.5×103/6.0×10-2 3.6×10–5/12×102 21×103×5×10-6/35×10-2

Convert into the required units. 46. 471 mg=? g 48.* 14×10-5 1=? µl 50. 42 µg=? mg

47. 18 ml=? 1 49. 0.000051 g=? µg 51. 831 µl=? ml

Exponents and Prefixes

52.* 54. 56. 58.

0.14×10–2ml=?µl 3.1×10-5mol/l=? mmol/l 0.031×105 nmol/1= ? µmol/l 0.15% (w/v)=? mg/l

53. 55.* 57.* 59.*

37

0.037 mmol/1=? µmol/1 17 mmol/1=? mol/m3 14pmol/µl=? mmol/ml 3.1×10-4 mg/ml=? % (w/v)

60. Complete Table 4.3, which shows the composition of phosphate-buffered saline. Use a calculator if necessary. Use the following atomic weights: Na=23, K=39, P=31, O=16, H=1, Cl=35.5, Mg=24, Ca=40 Table 4.3 Data for Question 60, Chapter 4.

61. What are the total phosphate and total chloride concentrations in this solution? Express your answers as both %(w/v) and mmol/1. 62. What are the concentrations of each of the metal ions in this solution? Express your answers as both %(w/v) and mmol/1. 63. Sodium pyruvate has a formula weight of 110. Calculate the weight required to make up 100 ml of a 50 mmol/1 solution. Complete Table 4.4 which shows the composition of various solutions made from this stock solution. Table 4.4 Data for Question 63, Chapter 4.

38

Easy Mathematics for Biologists

64. A bacterial cell culture has 1.0 ml removed and added to 9.0 ml of medium. 0.1 ml of this is removed and added to 9.9 ml of medium. 1 ml of this is removed and added to 9 ml of medium. 1 ml of this is removed and added to 9 ml of medium. The number of bacteria in 0.1 ml of the final dilution was found to be 23. What was the approximate number of bacteria in the original culture?

5

SOLVING EQUATIONS AND EVALUATING EXPRESSIONS

5.1 SOLVING EQUATIONS

You have already come across equations of two ratios in Chapter 3. An algebraic equation will usually show the relationship between a variable represented by a letter such as x or y and something else. To solve the equation means to find the value(s) of x that make(s) the equation true. Different equations that have the same solution are called equivalent equations. Example 5.1

In order to solve equations it is necessary to change them into simpler equivalent equations. You can do this by adding to or subtracting from both sides of the equation the same non-zero number. You can multiply or divide both sides of the equation by the same non-zero number. You may also substitute another expression for part of the original equation. Example 5.2

Example 5.3

39

40

Easy Mathematics for Biologists

Example 5.4

Example 5.5

Example 5.6

Note that when multiplying or dividing numbers in parentheses (brackets) you must multiply or divide all the numbers. Example 5.7

Solving Equations and Evaluating Expressions

41

and because 25=5×5, x=5 As I showed at the beginning of Chapter 3, if the numerators and the denominators of two ratios are represented by letters,

This gives a quicker way of simplifying proportions than multiplying both sides of the equation by both denominators. Example 5.8

Example 5.9

The solution of an equation of this kind is not always so simple: sometimes there are two possible values for x. Example 5.10

42

Easy Mathematics for Biologists

Subtracting 6x+6 from both sides x2+x-6=0 We now need a way of finding x. One way of doing this is to express the equation as two factors.

Equations where the unknown variable x is present to the power 2 (but not greater than 2) are known as quadratic equations. The last example was therefore finding the solution of a quadratic equation. In this example, it was possible to see how it could be factored into the form a×b=0, and then, in turn a and b are set equal to 0. However, many quadratic equations cannot be easily factored and another method using a formula has to be used. For an equation of the type ax2+bx+c=0, where a is a non-zero number, the solutions are given by:

means’ the square root of what is under the sign’. A square root of a number is one of its equal factors. =the number a such that a2=b

Solving Equations and Evaluating Expressions

43

It is not essential that you know how this formula is derived. However, you should be aware of its existence and be able to use it. Example 5.11

Note these are the same solutions as you would get by factoring.

5.2 EVALUATING EXPRESSIONS

To evaluate an expression, substitute the given values for the unknown variable. You should already be familiar with this for simple examples. Example 5.12

The circumference c of a circle is given by the formula c=πd where d is the diameter. What is the circumference of a circle with a diameter of 5 m? c=π(5)=15.7 m

Example 5.13

What is the radius of a circle with a circumference of 15.7 m? c=πd rearranges to d=c/π d=15.7/π=5 radius=d/2=5/2=2.5 m

Example 5.14

What is the area A of a circle of radius 3 m if the area is given by the formula A=πr2 where r is the radius? A=π(3)2=28.3 m2

44

Example 5.15

Easy Mathematics for Biologists

The lengths of the three sides of a right-angled triangle are related by the equation x2+y2=z2 where x, y, and z are the lengths of the sides, z being the side opposite the right angle. What is the length of side x when z=5 m and y=4 m? x2+42=52 x2+16=25 x2=25-16=9 so length of side x=3 m.

It is important when evaluating expressions that you are careful that you substitute values with the correct units. Example 5.16

If the amount of calcium in a 1.0 g sample of bread is 5.0 mg, what is the calcium concentration expressed as %(w/w)? %(w/w) = (weight of calcium/weight of bread) × 100. Direct substitution would give calcium concentration=(5/1)×100=500

This is wrong because the units do not match. The correct calculation is

Example 5.17

In the formula A=E c d A=the absorbance of a solution as measured in a spectrophotometer, E=the absorption coefficient of the solute c=the concentration of the solute d=the length of the light path through the solution What is the absorbance of a solution of 0.1 mmol/l p-nitrophenol if the absorption coefficient is 1.8×104 1 mol-1cm-1 and a light path of 10 mm is used?

Note that in this formula A has no units, so the units of E must be the inverse of the units of c×d, i.e. the units of E are litres/(mol×cm) since the units of c×d are (mol/1)×cm.

Solving Equations and Evaluating Expressions

Example 5.18

45

The equilibrium constant, Keq, for a reaction A+B→C is given by

where [A], [B], and [C] represent the concentrations of the reactants A and B and of the product C at equilibrium. Calculate the equilibrium constant if A=0.2 mol/l, B=0.5 mmol/l, and C=15 µmol/l.

5.3 PROBLEMS If you have just finished reading this chapter I suggest you now try the ‘pure mathematics’ examples marked‘*’. If you get these correct, try all of the applied examples. If you also get these correct, move on to the next chapter. If you do not get the right answers, re-read the explanations and try the rest of the problems 1– 24, and then repeat your attempts at the applied problems. Solve each of the following equations. 1. 3. 5. 7. 9.

3x+15=6 11x+5-3x-7=42 7(2x-3)=21 7x/3-1/5=11

11. 0.34x-1.65=11.71

2.* 4. 6.* 8.* 10.*

7-2x=11 7x-15=5-3x 4(11-3x)=2x+16 3x/5+1/4=11/20

12. 0.31x +0.05=17.1-0.003x

13.

14.*

15.

16.

17. 11/(x-4)=22/(x-1)

18.*

19.

20.

21. 23.

22.* 24.* (x-3)/5=0.5x/(x+0.5)

46

Easy Mathematics for Biologists

25. What is the concentration of a solution of NADH if the molar absorption coefficient=6.3×103 1 mol-1 cm-1 and the absorbance=0.155 in a cuvette with a light path length of 4 cm? 26. For many enzyme-catalysed reactions, where [S] represents the concentration of substrate (reactant), the rate of reaction at a particular concentration of substrate is given by:

Vmax is the maximum rate of reaction and Km is a constant. (i) (ii) (iii) (iv) (v)

Calculate v when [S]=0.2 mmol/l, Vmax=0.3 µmol/s and Km= 400 µmol/l. If the Km=0.4 mmol/l, find the ratio when [S]= 2.0 mmol/l. Find Km if and [S]=0.2 mmol/l. Find [S] if v=0.91×Vmax and Km=0.4 mmol/l. Complete Table 5.1 showing the relationship between and

27. The dissociation of a weak acid can be written as HA where HA represents the weak acid, H+ is a hydrogen ion, and A¯ is the ‘conjugate base’, i.e. the original molecule minus the hydrogen ion. For example, for acetic acid,

Dissociation is only slight, so at a particular temperature an equilibrium will be reached. The equilibrium constant for the dissociation (Ka), called the acid dissociation constant or ionisation constant is

Table 5.1 Data for Question 26(v), Chapter 5.

Solving Equations and Evaluating Expressions

47

Calculate the [H+] of a solution made by using 0.5 mol of acetic acid in 1 litre of water at 25°C. HINT: If x=[H+], then it also=[H3CCOO-], while [H3CCOOH]= the original acetic acid concentration minus x.

6

6.1

LOGARITHMS

LOGARITHMS AND EXPONENTS

Exponents were introduced in Chapter 4. In section 4.2 you will have seen that numbers can be written as a combination of a base and a raised numeral, the exponent. For example, 16=24 where the base is 2 and the exponent is 4. The logarithm (log) of a number has the same value as the exponent: the power to which a base must be raised so that it equals the given number. Example 6.1

log216=4 because 24=16 Read this as ‘the log to the base 2 of 16 equals 4 because 2 to the power 4 equals 16’. Likewise, log10100=2 because 102=100

In general, then, logb a=x when bx=a Since the value of a logarithm depends on the base, this must be specified or implied. ‘Logarithm’ is usually abbreviated to ‘log’ with the specified base as a subscript, e.g. log2. However, the most frequently encountered logs are those with base 10 (called common logs) and those with base e (called natural or Naperian logs: e being a number with a value of approximately 2.718. It is not necessary to understand how it is derived to use it.) A common log is usually written as ‘log’, and a natural log as ‘In’. This is how they appear on calculator keys. So, log2 a specifies the base 2 whereas log a implies log10a (a common log) and In a implies logea (a natural log) Note that any positive numbers can have logs, but zero and negative numbers cannot. What is log100? It means 10x=0 but there is no value of x for which this could be true: a positive or negative value of x gives a value greater than 0, and 100=1. What is log10(-10)? 10x=-10. Again, there is no value of x for which this could be true. You will remember from section 4.2 that when multiplying numbers expressed in exponential form, you add the exponents. Therefore the log of two exponential numbers (having the same base) multiplied together equals the sum of the logs of the numbers. 49

50

Easy Mathematics for Biologists

Example 6.2

The general rule is:

Similarly, when exponential numbers are divided, the exponents are subtracted. Therefore the log of one exponential number divided by another (having the same base) equals the difference between the logs of the numbers. Example 6.3

The general rule is:

At the beginning of this chapter you saw that

Thus the general rule is:

Example 6.4

So far I have used only examples where the logs are whole numbers, but there is no need for this restriction. Example 6.5 log 25=1.3979 because 25=101.3979 In the example above, 1.3079 is the log of 25, but also 25 is the antilog of 1.3979. The antilog of a number is the value obtained when the base is raised to a power equal to the number to be antilogged.

Logarithms

51

Example 6.6 log 105=5, and antilog 5=105 Note that this means the antilog of the log of a number is the number itself.

Two other rules follow: (1)

(2) means the nth root of a, i.e. the number which when multiplied by itself n times equals a, so it is an inverse power of a.

Example 6.7

6.2 MANIPULATING LOGS

Using the rules for logs explained above, it is possible to rearrange, simplify, or solve equations containing logs. The rules are restated here in various useful forms. For any positive non-zero number a and any positive non-zero number b, (1) (2) (3) (4) (5) (6) (7)

log(ab)=loga+logb a×b=antilog (loga+logb) log(a/b)=loga–logb a/b=antilog(loga–logb) logan=nloga

52

Easy Mathematics for Biologists

Example 6.8

Solve log2(x-4)=3 Taking antilogs of both sides antilog(log2(x-4))=antilog23 So x-4=23=8 x=12

Example 6.9

Express y=ax-c in natural logarithmic form Following rule (1) ln y=ln a+ln x-c Following rule (6) ln y=ln a-c ln x

Example 6.10

Using log 2=0.3 and log 3=0.48, calculate the following, without using a calculator.

Example 6.11

Express ln x in terms of log x. Remember that the natural log, ln, has the base e. If ln x=a, x=ea Using rule (6) log x=log(e)a =alog e so a=log x/log e=log x/0.434=2.303log x ln x=2.303log x

Example 6.12

Find n when 9=8n Using rule (6) log 8n=nlog 8 so n=log 9/log 8=1.057

You can check this is correct using your calculator if you have a key labelled ‘xy’. Press: 8, xy, 1.057, =. You may have to use a supplementary key to obtain xy,

Logarithms

53

e.g. the INV key, the second function key, or the shift key in which case the sequence would be, as an example, 8, INV, xy, 1.057, =. Example 6.13

Find n when n=(1.8)2.5/(2.7)3.2 Following rule (3),

Example 6.14

Solve log2(x-4)=3 antilog(log2(x-4)) = antilog23 x-4=23=8 x=12

Example 6.15

Note that if your calculator has an ‘x1/y’ button, you could calculate this directly. Press: 20, INV (if necessary), x1/y, 3, =. In Chapter 5, problem 27 involved the calculation of the [H+] in a solution of a weak acid with an acid dissociation constant represented by Ka. Because of the sizes of the values for these it is convenient to transform these numbers by taking logs, and inventing a term for these transformed numbers. Thus, the ‘pH’ of a solution is defined as the negative logarithm to the base 10 of its hydrogen ion concentration, i.e.

Similarly, the pKa of a weak acid is defined as the negative logarithm to the base 10 of its acid dissociation constant, i.e.

54

Easy Mathematics for Biologists

Water itself dissociates, so it also has a Ka value which at 25°C is 1.8×10-16 mol/l.

More correctly, but for most purposes you can think of a hydrogen ion being formed rather than a hydronium ion (H3O+), so

Pure water is 55.6 mol/l, ignoring the negligible reduction in concentration caused by ionisation, so

This is known as Kw, the ion product of water.

Using this relationship it is possible to calculate the pH of a solution of a base. Example 6.16

What is the pH of a 0.2 mol/l aqueous solution of NaOH at 25° C? NaOH is a strong base which completely dissociates. It will give rise to 0.2 mol/l OH-, so [OH-]=0.2 mol/l.

The Henderson-Hasselbalch equation expresses the relationship between pH, pKa, and the ratio of the concentrations of conjugate base and weak acid when a weak acid is neutralised by a strong base.

Logarithms

Example 6.17

55

Check that the answers obtained in problem 27 in Chapter 5 satisfy this equation.

The Henderson-Hasselbalch equation is particularly useful in calculating the amounts of acid and conjugate base required to make a buffer solution of a certain pH. Example 6.18

How could you prepare 1 litre of 0.1 mol/l sodium phosphate buffer pH 7.2, given solutions of 0.1 mol/l Na2HPO4 and 0.1 mol/l NaH2PO4? The pKa value for the dissociation

The proportion is therefore 2.51 volumes of 0.1 mol/l Na2HPO4 to l volume of 0.1 mol/l NaH2PO4.

56

Easy Mathematics for Biologists

litres of 0.1 mol/1 Na2HPO4 plus 0.285

So you need

litres of 0.1 mol/l NaH2PO4. Add them and mix. The equation can also be used to calculate the proportion of molecules of a weak acid or base that are ionised at a certain pH. Example 6.19

Glycine has the structure H3N-CH2-COOH. The pKa for the carboxyl group is 2.4. What proportion of the glycine molecules in a dilute aqueous solution have ionised carboxyl groups at pH 3.4?

Therefore at pH 3.4, there will be 10 molecules of glycine with an ionised carboxyl group for every one with the group un-ionised. Therefore the proportion ionised=10 ionised/11 total=0.91 or 91%.

6.3 PROBLEMS

If you have just finished reading this chapter I suggest you now try the ‘pure mathematics’ examples marked ‘*’. If you get these correct, try all of the applied examples. If you also get these correct, move on to the next chapter. If you do not get the right answers, re-read the explanations and try the rest of the problems 1– 54, and then repeat your attempts at the applied problems. Find the values of the following without using a calculator. 1. 3. 5. 7. 9. 11.

161/2 log 100 ln 1 log 0.0001 log10-7 log2 16

13. 15. log255

2.* 4.* 6. 8.* 10. 12.*

272/3 log104 ln e5 ln (e2)3 log5125

14. log832 16.* log328 (Hint: think in terms of base 2)

Logarithms

57

Using log 2=0.30 and log 3=0.48, and without using a calculator, find the values of the following. 17. 19. 21. 23. 25.

log 4 log 0.6 log 200 log (6×10-3)

18. 20.* 22. 24.* 26.*

log 36 log 1.5 log 5 log(3×104) log 0.04

28. 30.* 32. 34.* 36. 38. 40.* 42. 44.* 46. 48.* 50. 52.

103x-1=5 10(x2)=16 log3x=4 log5x=-3

Solve the following equations. 27. 29. 31. 33. 35. 37. 39. 41. 43. 45. 47. 49. 51.*

2x+1=8 9x+3=3x log4x=2 log2x=7 log4x=-2 10

–3

log x=10 logx8=3 logx64=-2 log x+log 5=-1 log(x+1)+log 5=4 log(x+1) + log(x-1)=3 log(x+1) + log(x-4)=log 6

log10x=32 logx169=2 log x-log 0.01=0.1 log(x-2)-log 4=3 log(x+1)-log(x-3)=7 log(2x+1)-logx =log(x+2)

53.* Express in natural logarithmic form the equation y=axb 54. Express in exponential form the equation log y=alog x+b. 55. You can set a spectrophotometer to read either transmittance (T) or absorbance (A). Percentage transmittance is the percentage of the incident light that passes through the sample. Absorbance is related to transmittance by the equation: A=-log T. (i) Complete Table 6.1. (ii) What is the transmittance of a solution with an absorbance of 0.5? 56. Calculate the pH values of solutions with the following H+ concentrations. (i) 0.1 mol/l (ii) 0.0001 mol/l (iii) 5×10-3 mol/l (iv) 8×10-11 mol/l (v) 6×10-3 mmol/l 57. Calculate the pH values of solutions with the following OH- concentrations. (i) 0.1 mol/l (ii) 0.001 mol/l

58

Easy Mathematics for Biologists

Table 6.1 Values for Question 55, Chapter 6.

(iii) 3×10-3 mol/l (iv) 1.7×10-5 mmol/l (v) 11×10-8 mol/l 58. Calculate the H+ and OH- concentrations of solutions with the following pH values. (i) 2.1 (ii) 8.0 (iii) 3.6 (iv) 11.4 59. Ammonia is a weak base that in aqueous solution forms its conjugate acid NH4+ by reaction with water:

60.

61. 62.

63.

64.

The pKa for the NH4+ is 9.25. What will be the pH of a 0.01 mol/l solution of NH3? (Hint: think of the reverse reaction.) The Ka of the conjugate acid of the base ethylamine is 1.58×10-11 mol/l. Which base, ammonia or ethylamine, is stronger, i.e. produces a solution with the higher pH for the same concentration? What is the pH of a solution made by adding 100 ml of 0.05 mol/l acetic acid to 200 ml of 0.01 mol/l sodium acetate? (pKa of acetic acid is 4.74.) If the amino acid alanine has pKa values of 2.3 for the carboxyl group and 9.7 for the amino group, what proportion of the carboxyl and amino groups will be ionised at pH 7.4? If the pH of blood is 7.4 and the total concentration of inorganic phosphate (consisting of PO43-, HPO42-, H2PO4- and H3PO4) is 1 mmol/l, what are the approximate concentrations of the ions PO43-, HPO42-,and H2PO4-. (Use pKa values of 2.14, 6.86 and 12.4.) The equilibrium constant for a reaction is related to the standard free energy change ∆G° by the equation:

where R=8.3 J K-1 mol-1 and T is temperature (kelvin). Calculate ∆G° for a reaction when the temperature is 37° C and Keq=0.50.

7

STRAIGHT-LINE GRAPHS: CALIBRATION CURVES AND LINEAR RATES OF CHANGE

7.1 PROPORTIONAL RELATIONSHIPS

In previous chapters you have come across many examples where one number is related to another in a very simple way. For example, the circumference of a circle is π times the diameter and the absorbance of a solution is a constant times the concentration. These relationships can be expressed as: c=πd, where c is the circumference, and d is the diameter, and A=kc, where A is the absorbance, k is a constant and c is the concentration. Note that each of these can be rearranged as a proportion, i.e. so that the value of one variable is proportional to the value of the other:

In experiments we are often trying to find out the relationship between two variables, i.e. to answer questions of the type. ‘How does the value of one variable, y, change when the value of the other variable, x, is changed’. An experiment to answer this question is likely to produce a set of data consisting of values of x, each of which has a corresponding value of y. One way of analysing this data to show the relationship between x and y is to plot them on a graph in which the horizontal distance represents the value of x, and the vertical distance the value of y. Since each experimental value of x has an associated value of y, it can be represented by a point whose co-ordinates are those values. Any point, with the coordinates xn and yn, can be written as Pn(xn, yn). Example 7.1

The data point P1 where x=2 and y=3 is plotted as shown in Graph 7.1. Note that there are x and y axes having scales starting at 0 and that each unit of distance along the axis represents the same increase in value. Remember that the horizontal x coordinate is always given first, so the point can be labelled P1(2, 3). 59

60

Easy Mathematics for Biologists

Graph 7.1

Example 7.2

From an experiment the results shown in Table 7.1 were obtained. The data can be plotted on a graph as shown in Graph 7.2.

Each point on this graph represents a pair of values of x and y. Note that in this example, all the points lie on a straight line which passes through the origin (the point (0, 0)). Taking the point (0.100, 0.6) you can see that the value of y is 6 times the value of x (0.6=6× 0.100). If you look at any other point, you will see that this is still true. Check for yourself using the original data points and then some intermediate points. Since the value of y is always 6 times the value of x, the relationship can be written as: y=6x which is an equation of the form y=ax where a is a constant, and the value of y is proportional to the value of x, i.e. . On this same graph,

is the steepness of the straight line, the ratio of the

vertical change to the horizontal change. This is usually known as the gradient or Table 7.1 Data for Example 7.2.

Straight-Line Graphs

61

Graph 7.2

slope of the line. If you start at P1 (x1, y1) and move along the line to P2(x2, y2), you will have moved vertically by y2-y1 and horizontally by x2-x1, so the gradient=(y2-y1)/(x2-x1). (This is also sometimes written as ∆y/∆x where delta y and delta x denote the change in y and x. The gradient is therefore a measure of how many units y changes for a change of 1 unit in x. Example 7.3

Find the value of the gradient in Graph 7.3. You could obtain the gradient by choosing two points e.g. (0.100, 0.60) and (0.050, 0.30) and calculating the gradient as:

Note that it does not matter which point is P1 and which is P2:

Example 7.4

Find the gradient of the straight line passing through the points P1(1, 3) and P2(3, 9).

62

Easy Mathematics for Biologists

Graph 7.3

7.2 EQUATIONS OF STRAIGHT LINES

Look at Graph 7.4. How does y change for a change in x? Choose any two points e.g. (0, 2) and (5, 5). The ratio . As this is true for any two points, the change in y is proportional to the change in x, but in this case y is not proportional to x. This is because the line does not pass through the origin, the point where both x and y are zero. The line cannot therefore be represented by the equation y=0.6x, because, for example, if you substitute

Graph 7.4

Straight-Line Graphs

63

the coordinates of the point (5, 5) in the equation you get 5=0.6×5=3 which is nonsense. If you look again at the graph you can see that when x=0, y=2, so the correct equation is y=0.6x+2. In general, any straight line can be represented by an equation which can be expressed in the form y=ax+b where a and b are constants, a is the gradient and b, the value of y when x=0, is called the y-intercept. Both a and b can have positive, zero, or negative values. Example 7.5

-3x+6y=9 Simplifying gives 2y=x+3 -x+2y=3 y=0.5x+1.5

In this case, the gradient is 0.5 and the y-intercept is 1.5. In Graph 7.5, the y-intercept b=-1 as the line intercepts the y-axis at the point (0, -1). The gradient is (0-(-1))/(2- 0)= =0.5. Therefore the equation of this line must be y=0.5x-1. In Graph 7.6, the y-intercept is the point (0, 5) and the slope is (0-5)/ (5-0)=-5/ 5=-1, so the equation of this line must be y=-x+5. In Graph 7.7 the line is parallel to the x-axis so the value of y does not change whatever the value of x, so the gradient=0. The value of y is said to be independent of x. Since the y-intercept is the point (0, 2) the equation of the line must be y=2. In Graph 7.8, the line is parallel to the y-axis and there is no y-intercept. The line represent a set of values of y which have no defined relationship to x, so the gradient is undefined and the equation of this line is x=-2.

Graph 7.5

64

Easy Mathematics for Biologists

Graph 7.6

Graph 7.7 It is possible to determine the equation of a straight line given either • the coordinates of two points on it, or • the coordinates of one point plus the gradient, or • the gradient and the y-intercept. Example 7.6

Determine the equation of the straight line passing through the points (-3, 7) and (5, 11).

Straight-Line Graphs

65

Graph 7.8

Using the values of y and x at one point 11=0.5(5)+b 11=2.5+b b=8.5 Therefore y=0.5x+8.5. Example 7.7

Determine the equation of the straight line passing through the point (3, 4) with a gradient of -5. The equation will be of the form y=ax+b, where a=-5 substituting the values of y and x 4=-5(3)+b 4=-15+b b=19 so y=-5x+19

Example 7.8

Determine the equation of the straight line with a gradient of -3 and y-intercept of -4. y=ax+b where a is the gradient and b the y-intercept, so y=-3x-4

Given the equation of a straight line you can determine the coordinates of two points by substituting values for x. Example 7.9

y=-5x+20 If x=1, then y=-5+20=15 so one point is (1, 15) If x=2, then y=-10+20=10 so another point is (2, 10).

66

Easy Mathematics for Biologists

Using these coordinates you could draw a graph with the straight line passing through these points. An alternative is to find the x and y intercepts. When x=0, y=20 so the y-intercept is (0, 20) When y=0, 5x=20, x=20/5=4 so the x-intercept is (4, 0).

7.3

EQUATIONS OF STRAIGHT LINES AND STRAIGHT-LINE GRAPHS IN PRACTICE

If you carry out an experiment in which you change the value of x and determine the corresponding value of y, you can plot the values of x and y on a graph. You should make sure you do the following six things: (a)

(b)

(c) (d) (e)

(f)

Choose appropriate scales and mark them on the axes. The scales will normally be linear; for example 1 cm along the axis might represent a change of 1 minute wherever you start from on the axis. You should normally include the origin on your graph. If you do not, you should make it obvious that this is the case, perhaps by having a break in the axis. Label the axes to indicate what variables are being plotted. The units on the x-axis are those of the independent variable. This is the one whose values may have been chosen, or do not depend on the other variable. The dependent variable (the one whose value is determined by the other variable) is plotted on the y-axis. Label the axes with appropriate units. Mark the points clearly. Decide whether or not the points lie on a straight line. This is not always easy. Because of experimental error or natural variation the data points are unlikely to lie exactly on a straight line. Also, in some cases, there may be a proportional relationship between the variables for only part of the range of values of x. These points are illustrated in Graphs 7.9 and 7.10. Graph 7.9 shows a situation where y is proportional to x only over part of the range of x-values. In Graph 7.10 there is some doubt whether a straight-line relationship really exists. Graphs of curves are dealt with in Chapter 8. If you are convinced that there is a straight-line relationship between x and y you need to draw the line which best fits the points. You can do this by eye, trying to balance points above and below the line, or you can use a statistical method (linear regression) which is available on many calculators and in

Straight-Line Graphs

67

Graph 7.9

Graph 7.10 computer statistical or data presentation packages such as MINITAB. (Statistical methods can be used to determine the goodness-of-fit of the data to a straight-line relationship.) Having drawn the line you can determine the gradient and y-intercept, and hence the equation of the line. You can interpret a straight-line graph as follows: (a)

The variable y is related to x such that any increase in x is associated with a proportional increase in y. The gradient of the line gives the ratio between the change in y and change in x.

68

(b) (c)

Easy Mathematics for Biologists

If the line passes through the origin, y is proportional to x. If the line does not pass through the origin, when x=0, y has a non-zero value.

You may be able to predict values of y for another set of values of x, or use measured values of y to determine x. Example 7.10

The results shown in Table 7.2 were obtained when the intensity of the light emitted from a set of solutions containing different known concentrations of Ca2+ (a set of ‘standards’) was measured in a flame photometer.

Table 7.2 Data for Example 7.10.

If the readings for these standards are plotted against their Ca2+ concentrations (µg/ml), a straight-line graph is obtained which passes through the origin and has a slope of 0.5 (see Graph 7.11).

Graph 7.11

Straight-Line Graphs

69

The equation is therefore reading=0.5(concentration(µg/ml)). This indicates that the reading is proportional to the Ca2+ concentration over the range used. If readings are obtained from other solutions (samples with unknown Ca2+ concentrations), their concentrations can be calculated by dividing the reading by the gradient. reading=0.5 (concentration) so sample reading/0.5=concentration (µg/ml). This example illustrates the use of a ‘calibration curve’ which is a general term for the graph obtained when the readings from an instrument are plotted against the known values of another variable. In many experiments time is the independent variable (x), in which case the gradient of the straight-line graph represents the rate of change of the other variable. Suppose you are measuring the amount of product obtained from a reaction at various times after starting the reaction. When you plot the amount of product against time, you find the points lie on a straight line passing through the origin. You are therefore able to conclude that product accumulation is proportional to time over the range measured. The gradient gives the rate of reaction, i.e. the rate at which product is made. If you repeated this experiment at a higher temperature, you might find that the gradient increased. You could conclude that the rate of reaction depends on the temperature. Another example would be if you were measuring the rates of reaction using different concentrations of enzyme. For each enzyme concentration you would plot a graph of product accumulated against time, and obtain the gradient of the straight line. These gradients represent the rates of reaction at different enzyme concentrations. When you plot these rates of reaction against enzyme concentration you find the points lie on a straight line passing through the origin. You are therefore able to conclude that under the conditions used, the rate of reaction is proportional to the enzyme concentration.

7.4 PROBLEMS

If you have just finished reading this chapter I suggest you now try the ‘pure mathematics’ examples marked ‘*’. If you get these correct, try all of the applied

70

Easy Mathematics for Biologists

examples. If you also get these correct, move on to the next chapter. If you do not get the right answers, re-read the explanations and try the rest of the problems 1– 16, and then repeat your attempts at the applied problems. Find the gradient of the straight line passing through the points whose coordinates are given. 1. (2, 3), (5, 6) 3. (0, -5), (6, 5)

2.* (-1, 3), (-5, -6) 4. (0.25, 4), (4, -1.5)

Find the equation of the straight line passing through the points whose coordinates are given. 5. (3, 2), (6, 5) 7. (2, 3), (-6, 5)

6.* (3, 1), (5, 6) 8.* (-1, -1), (2, 8)

Find the equation of the straight line passing through the point P1 with the given gradient a. 9. P1(-1, 2), a=2 11.

10. P1(-1, -3), a=-1 12.*

Graph the following equations: 13. y=2x+1 15. 3–2y=4x

14. y=-2x-3 16.* 4y-3x=9

17. A 0.25 mmol/1 stock solution of p-nitrophenol in buffer at pH 10.0 was diluted in buffer to prepare a set of solutions of different concentration. The absorbance at 410 nm of each of these solutions was measured in a spectrophotometer using a 1 cm light path. The results are given in Table 7.3. Calculate the molar absorption coefficient for p-nitrophenol if A=Ecd where A=absorbance, E=molar absorption coefficient, c=concentration, and d=light path length. Table 7.3 Data for Question 17, Chapter 7.

Straight-Line Graphs

71

18. Using the data in 17 above, calculate the concentrations of the unknown solutions of p-nitrophenol which when diluted as indicated, gave the absorbances shown in Table 7.4. 19. In an assay for the enzyme lumpase in saliva, the enzyme is incubated with the substrate, L-LUMPA, and the rate of formation of product, BITTA, is measured by taking samples at times during the incubation and adding concentrated HCl which converts BITTA to a coloured substance, MUDDY. The intensity of the colour can be measured spectrophotometrically and is proportional to the concentration of MUDDY. For every molecule of L-LUMPA used, one molecule of MUDDY is produced. An experiment was performed as follows: 1.0 ml saliva+9.0 ml buffer-L-LUMPA solution were incubated at 37° C. 1.0 ml samples were taken at set times, 2.0 ml cone. HCl added and the absorbance measured against a suitable blank. A 1.0 cm light path was used. The results for this are given in Table 7.5, and those for a calibration curve are given in Table 7.6. Calculate: (i) (ii) (iii)

The rate of reaction in units of absorbance change minute-1. The rate of reaction in units of µmoles L-LUMPA converted minute-1. The activity of the enzyme in units of µmol minute-1 ml-1 saliva.

Table 7.4 Data for Question 18, Chapter 7.

Table 7.5 Data for Question 19, Chapter 7.

72

Easy Mathematics for Biologists

Table 7.6 Calibration Curve Data for Question 19, Chapter 7.

Recently an artificial substrate (JOBBO) for the enzyme has been made which gives a red-coloured product without the need for the addition of cone. HCl. This product has a molar absorption coefficient of 5,000 1 mol-1cm-1. In an assay with the same saliva sample as before, when 1.0 ml saliva was incubated with 9.0 ml buffer-JOBBO solution at 37°C, the absorbance increased linearly with time from a value of 0 at time=0 min, to 0.5 at time=10 min. (iv) Calculate the activity of the enzyme in the sample. 20. You obtain the results shown in Table 7.7 in an experiment to determine the effect of substrate concentration on the rate of an enzyme-catalysed reaction. Plot a graph of rate of reaction against substrate concentration. What can you say about the relationship between the two variables?

Table 7.7 Data for Question 20, Chapter 7.

8 NON-LINEAR RATES OF CHANGE: GRAPHS, TRANSFORMATIONS AND RATES

8.1

GRAPHS THAT ARE NOT STRAIGHT LINES, AND THEIR TRANSFORMATION

Question 20 in the previous chapter had data that when plotted directly gives a curve rather than a straight line. You will often find this to be the case, because the relationship between many variables is not one of direct proportionality. If the rate of a chemical reaction is proportional to the concentration of the reactant A, for the reaction A→B, the rate of reaction=k[A] where k is a constant (called the rate constant). Also, the rate of reaction = the rate of disappearance of A which can be represented by the expression where the ‘d’ stands for ‘a small change in’, and the minus sign is present because [A] is falling. Example 8.1

Find the rate of reaction at 2.5 and 5.0 minutes, using the data in Table 8.1. If [A] is plotted against time, the Graph 8.1 is obtained.

You can see that it is not easy to get accurate values for the rates of reaction at 2.5 and 5.0 minutes because these are the gradients at these times, and the graph is a curve rather than a straight line. This implies that the rate of reaction is constantly changing, and this is of course because the concentration of A, which determines the rate, is constantly falling. One way of calculating the Table 8.1 Data for Example 8.1.

73

74

Easy Mathematics for Biologists

Graph 8.1

Graph 8.2 gradient more easily is to transform the data mathematically so that when replotted the graph becomes a straight line. In this example you calculate the natural logs of the values for [A]. Graph 8.2 shows a plot of In [A] against time. You can see that this is a straight line. Therefore its equation must be of the form y=ax+b. In this case, y=ln[A], x=time, and b is the y-intercept which=4.38. The gradient is a, which you can calculate in the usual way from

Non-Linear Rates of Changes

75

two points, and which equals -0.429. Therefore the equation of this straight line is: ln[A]=-0.429 (time)+4.38 The gradient a is also equal to the rate constant, k. The rates of reaction at 2.5 and 5.0 minutes can be obtained by first finding [A] at these times. So, at 2.5 minutes, In [A]=-0.429(2.5)+4.38=3.31 So [A]=antiln(3.31)=27.3 mmol/l and since rate of reaction=k[A] rate of reaction=-0.429(27.3) =-11.7 mmoll-1 minute-1. Similarly, at 5 minutes, ln[A]=-0.429(5)+4.38=2.26 so [A]=antiln(2.26)=9.54 mmol/l and rate of reaction=-0.429(9.54) =-4.09 mmol 1-1 minute-1. Note that the rates of reaction are negative because they represent the rate of fall in concentration of A. The example above shows how it may be possible to obtain an equation relating two variables if one or both variables can be transformed so that when the transformed data is plotted, a straight line is obtained. Therefore you might expect that several shapes of curve can be described by relatively simple equations. Graphs 8.3 to 8.8 are examples. Graph 8.3 is of the equation y=beax where both a and b are positive (in this case, 0.5 and 2 respectively). This is an exponential equation. Graph 8.4 is of the equation y=beax where a is negative and b is positive (in this case, -0.5 and 5, respectively). This is also an exponential equation. Graph 8.5 is of the equation y=a(lnx) where a is positive (in this case 0.5). This is a logarithmic equation. Graph 8.6 is of the equation y=bxa where both a and b are positive (in this case, 0.5 and 2 respectively). Graph 8.7 is of the equation y=ax2 where a is positive (in this case 0.5). Graph 8.8 is of the equation where a and b are positive (in this case, 0.75 and 1.25 respectively). In each of these cases, it is possible to obtain an equation relating x and y, and the values of the constants a and b, by transforming the data in the following ways:

76

Easy Mathematics for Biologists

Graph 8.3

Graph 8.4

• For equations of the form y=beax (Graphs 8.3 and 8.4)

If ln y is plotted against x, the gradient=a and the intercept on the ln y axis=ln b.

Non-Linear Rates of Changes

Graph 8.5

Graph 8.6

• For equations of the form y=a(ln x) (Graph 8.5) If y is plotted against ln x, the gradient = a. • For equations of the form y=bxa (Graph 8.6)

77

78

Easy Mathematics for Biologists

Graph 8.7

Graph 8.8

If ln y is plotted against ln x, the gradient=a and the intercept on the ln y axis is ln b. • For equations of the form y=ax2 (Graph 8.7) If y is plotted against x2, the gradient=a. • For equations of the form

(Graph 8.8) there are several possibilities.

79

Non-Linear Rates of Changes

(i) Invert the equation:

This is the same as

If

is plotted against

intercept=

, the gradient= , and the

intercept =

. Also, the

-

.

(ii) Invert the equation:

Multiply by x

If

is plotted against x, the gradient is

and the

intercept is .

(iii) Multiply by the denominator, then divide by x:

If y is plotted against , the gradient is -b and the y-intercept is a.

8.2 RATES OF CHANGE

In the previous section you have seen how variables may be related in such a way that the rate of change of y with a change in x is itself constantly changing. In example 8.1 where the rate of change was the rate of a reaction =the rate of disappearance of A, I introduced the expression -d[A]/dt where the ‘d’ stands for

80

Easy Mathematics for Biologists

‘a small change in’, and the minus sign is present because [A] is falling. The term is a general term for the change in y relative to the change in x and is therefore the gradient of the graph of y against x for any shape of graph. The value of

at

any particular value of x is given by the ‘differential form’ of the equation of the graph of y against x. The latter can be called the ‘integrated form’ of the equation, and I gave examples of these in Section 8.1. (An explanation of the derivation of the differentiated forms is outside the scope of this book. You would need to read an ‘A’ level or higher level textbook to find out about it. However, you can use the differentiated forms of the equations in calculations without understanding their origins.) The differential forms of the equations are listed in Table 8.2. , the rate of change of By using these equations you can therefore calculate y with x, for any data whose points form a line which can be described by one of the listed integrated forms. Given a set of data values of x and y, you should do the following. (i) Plot y against x. If this gives an obvious straight line, you can determine the gradient and the intercept directly, and hence find , the equation of the line, and therefore the relationship between y and x. (ii) If the plot does not give an obvious straight line, see whether it more closely resembles any of the types of curve discussed above. If so, try transforming the data appropriately and replot. If you now obtain a straight line, you can determine the gradient and intercept, find the equation of the line, and find from the differentiated form of the equation. Example 8.2

The number of bacteria in a culture was measured at 10 minute intervals over one hour. The results are shown in Table 8.3. (i) What is the relationship between number of bacteria and time? (ii) Calculate the growth rate at 15 minutes.

Table 8.2 Integrated and Differentiated Forms of Equations.

Non-Linear Rates of Changes

81

Table 8.3 Data for Example 8.2.

A plot of number of bacteria/ml against time gives an upward curving plot that is obviously not a straight line, but suggests an exponential relationship, i.e. if the num ber of bacteria/ml at time t=Nt, then Nt=beat where a and b are constants. A plot of ln Nt against t gives a straight line with a gradient of 0.0345 min-1 and intercept of 10.3. Therefore ln Nt=10.3+0.0345t antiln (ln Nt)=antiln (10.3+0.0345t) =antiln (10.3)×antiln(0.0345t) Nt=3.01×104×e0.0345t Also, since at time 15 minutes, Nt=3.01×104×e0.0345(15) =3.01×104×e0.518 =3.01×104×1.68 =5.05×104 so growth rate =0.0345×5.05×104 =1742 bacteria/minute at 15 minutes Example 83

The rate of a reaction was measured at different temperatures. From the results shown in Table 8.4, determine the relationship between rate of reaction and temperature.

Table 8.4 Data for Example 8.3.

82

Easy Mathematics for Biologists

Graph 8.9 shows the rate plotted against the temperature. The points suggest an upward curve rather than a straight line. This suggests y and x are likely to be related according to an equation of the form y=beax, and therefore you should try plotting ln rate against temperature. This is shown in Graph 8.10. You can see the points appear to lie on a straight line with gradient 0.024, suggesting the equation: ln rate =0.024 (temperature)+ln b, or rate=be0.024(temp) However, if you plot ln rate against

you obtain Graph 8.11.

Again, the points appear to fit a straight line, this time with a gradient of 2.15×103, suggesting the equation:

This illustrates a potential problem in that more than one transformation may give an apparently satisfactory fit of a curve to the data. You need to be aware of this possibility. It may be possible to eliminate one equation by thinking about what values are obtained at the extremes. ln the example, using the first equation, when temperature is 0 K, rate=b which is non-zero.

Graph 8.9

Non-Linear Rates of Changes

83

Graph 8.10

Graph 8.11

This cannot be true, though, since at absolute zero the molecules would be motionless and could not react. Using the second equation, you obtain rate=0 at 0 K, which is sensible. Transforming data into a logarithmic form is also often useful when the values for the independent variable span several orders of magnitude. By taking the log of these values it is easier to plot a graph which shows the effective range of values

84

Easy Mathematics for Biologists

over which the dependent variable changes significantly. This is particularly applicable in physiology or pharmacology where you may be interested in the relationship between the concentration of a neurotransmitter or the dose of a drug and the response of a tissue.

8.3 PROBLEMS

If you have just finished reading this chapter I suggest you now try the ‘pure mathematics’ examples marked‘*’. If you get these correct, try all of the applied examples. If you also get these correct, congratulations! If you do not get the right answers, re-read the explanations and try the rest of the problems 1–9, and then repeat your attempts at the applied problems. Sketch graphs of the following equations and of their differentiated forms. Find the values of at x=2. 1. y=ex 4. y=e-x 7. y=ln x2

2. y=ex/3 5. y=ln x 8.*

3.* y=2e3/x 6.* y=10(ln x) 9. y=5x0.5

10. In an experiment to measure the effect of histamine on the contraction of isolated guinea-pig ileum, the extent of tissue contraction was measured at 11 different concentrations of histamine. The results are shown in Table 8.5. Calculate the histamine concentration which gives half the maximum effect. Table 8.5 Data for Question 10, Chapter 8.

Non-Linear Rates of Changes

85

11. A hormone H binds reversibly to its receptor R on cell membranes:

The equilibrium constant for the dissociation of hormone-receptor complex into free hormone and receptor is Kd. The data in Table 8.6 gives values for the amount of hormone bound by receptor at different hormone concentrations. Calculate the total number of receptors and the Kd. 12. Radioactivity arises from atoms which have unstable nuclei. These nuclei disintegrate spontaneously, emitting ionising radiation. Radioactivity is quantified by the rate of disintegration. For example, a sample of a radioactive substance can be said to have an activity of 5×106 disintegrations per second, meaning that in the sample, this number of nuclei disintegrate every second. Obviously, if nuclei are disintegrating, the activity of a sample must decrease over time as there are fewer nuclei left to disintegrate. The time it takes for the activity to fall to half of its original activity is called the half-life, t1/2. This is a characteristic of each radioisotope, i.e. each type of radioactive atom. Calculate the half life of a radioisotope X, and the activity after 2 weeks from time 0, given the data in Table 8.7. 13. For a simple reaction the rate of reaction=k[X] where k is called the rate constant. The value of k depends on the temperature and the relationship is given by k=Ae-Ea/RT where A is a constant depending on the nature of the reaction, R is the gas constant (= 8.3 joules mol-1K-1) and Ea is called the activation energy of the reaction. T is the absolute temperature (K). Calculate Ea, for a reaction, given the values of the rates of reaction at different temperatures shown in Table 8.8. 14. The data in Table 8.9 relate the average basal metabolic rates of several species of mammal to the average body masses of these mammals. Derive an equation that describes the apparent relationship between these two variables. 15. Using the data from question 20 in Chapter 7, find the relationship between rate and substrate concentration, and the value of the constants, by using three different transformations of the equation:

86

Easy Mathematics for Biologists

Table 8.6 Data for Question 11, Chapter 8.

Table 8.7 Data for Question 12, Chapter 8.

Table 8.8 Data for Question 13, Chapter 8.

Table 8.9 Data for Question 14, Chapter 8.

ANSWERS TO PROBLEMS CHAPTER 2

1. 1 3. 5. 7. 9.

2. 4. 6. 8. 10.

11. 30.66 12. 4.14 13. 1.04 14. 26.95 15. 169.638 16. 1.225 17. 3.45 18. 13.16 19. 5.45 20. 23.75 21. 0.63 22. 6.9 23. 3.51 24. 33.3% 25. 75% 26. 77.8% 27. 88.3% 28. 60 29. 28 30. 80 31. 25.1% 32. 0.45 g of protein; 0.21 g of nucleic acid; 0.24 g 33. See Table 9.1 34. 21.8%; ; 0.218 35. (i) 3 g of sucrose made up to 100 ml with solvent (ii) 15 g of sucrose made up to 500 ml Table 9.1 Answer to Question 32, Chapter 2.

87

88

(iii) (iv) (v) 36. (i) (ii) (iii)

Easy Mathematics for Biologists

25 g of sucrose plus 475 g solvent 2.5 ml of ethanol plus 497.5 ml solvent, 2 g of ethanol made up to 20 ml with solvent 0.15 g 0.15 g 0.25 g

CHAPTER 3

1. 3. 5. 7. 9. 11. 13. 14. 15. 16. 17. 18.

2. 4. 6. 30 ml/1 or 0.03 1/1 8. 10. 49 12.

12 32 2 50 000 cm or 500 m or 0.5 km 1:500 000 4.541 400 See Table 9.2 (i) 275.5 g (ii) 275.5 g (iii) 27.55 g (iv) 27.55%(w/v) (v) 0.0374%(w/v) (vi) 0.00226 mol/1 (vii) 0.181 1 or 181 ml (viii) 0.73 ml (ix) and (x) See Table 9.3 19. Using 0.5 mol/1 stock solution and water, (i) 0.4 ml stock+9.6 ml water (ii) 0.08 ml stock+9.92 ml water, or, 1 ml stock+4 ml water, then 0.4 ml of this+9.6 ml water. Table 9.2 Answers to Question 17, Chapter 3.

Answers

89

Table 9.3 Answers for Questions 17 (ix) and (x), Chapter 3.

(iii) (iv) (v) (vi) (vii) (viii)

1 ml stock+4 ml water, then 0.1 ml of this+0.9 ml water, 1 ml stock+4 ml water, then 0.3 ml of this+0.7 ml water, 0.11 ml stock+9.89 ml water. 1 ml stock+8 ml water, then 0.15 ml of this+4.85 ml water, 1 ml stock+17 ml water, then of 0.08 ml of this+1.92 ml of water. 1 ml of stock+17 ml of water, then 0.3 ml of this+0.7 ml of water. The answers given below for (ix) and (x) are not the only possible correct answers. Generally you should be trying to be practical and economical. (ix) 0.4 ml of stock+9.6 ml of water; then of this, 1.0, 0.75, 0.50, 0.25 and 0 ml with 0, 0.25, 0.50, 0.75 and 1.0 ml water respectively. (x) 1 ml stock+44 ml water; then of this, 10, 7.5, 5.0, 2.5 and 0 ml with 0, 2.5, 5.0, 7.5 and 10 ml of water respectively. 20. 120 ml/minute

CHAPTER 4

1. 8

2. 1

3.

4.

5.

6. 2

7.

8.

9. 11. a

10. a5 12.

90

Easy Mathematics for Biologists

a2 a-1 or 1/a a-2 or 1/a2 10 0.1 1.73×1010 3.73×10–4 3.51×10-4 0.000376 0.015 61 109 -3.04×10-2 -4.35×10-2 7.5×104 3×10-8 0.3 0.018 1 51 µg 0.831 ml 37 µmol/1 17 mol/m3 1.4×10-5 mmol/ml 3.1×10–5% (w/v) See Table 9.4 Total phosphate is 0.092%; 9.6 mmol/1. 62. Na+ is 0.35%; 153 mmol/1. Mg2+ is 1.2×10-3 %; 0.49 mmol/1. Ca2+ is 3.5×10–3 %; 0.88 mmol/1.

13. 15. 17. 19. 21. 23. 25. 27. 29. 31. 33. 35. 37. 39. 41. 43. 45. 47. 49. 51. 53. 55. 57. 59. 60. 61.

14. a-5 or 1/a5 16. a6 18. a6 20. 0.1 22. 3.5×10-5 24. 3.184576×103 26. 1.6×102 28. 0.000315 30. 150 32. 0.04786 34. 1.093×104 36. 2.89×104 38. 4.495×10–2 40. 270 or 2.7×102 42. 3.96×10-2 44. 3×10-2 46. 0.471 g 48. 140 µ1 50. 0.042 mg 52. 1.4 µ1 54. 0.031 mmol/1 56. 3.1 µmol/1 58. 1.5×103 mg/1

Chloride is 0.50%; 142 mmol/1. K+ is 0.016%; 4.18 mmol/1.

Table 9.4 Answer for Question 60, Chapter 4.

91

Answers

Table 9.5 Answers to Question 63, Chapter 4.

63. 0.55 g. See Table 9.5 for other answers. 64. 2.3×107 bacteria/ml.

CHAPTER 5

1. 4. 7. 10. 13. 16. 19. 22. 25. 26.

-3 2 4.8 -5.96 -19.5 5 -2.57 -2 and 1 6.15 µmol/l (i) 0.1 µmol/sec (ii) 0.83 (iii) 0.8 mmol/l (iv) 4.0 mmol/l (v) See Table 9.6 27. 2.99×10–3 mol/l

2. 5. 8. 11. 14. 17. 20. 23.

-2 3 0.5 39.3 14.67 7 -3.8 3 and -2

3. 6. 9. 12. 15. 18. 21. 24.

5.5 2 4.167 54.47 6 1.4 3 and -1 5.284 and -0.284

92

Easy Mathematics for Biologists

Table 9.6 Answers to Question 26 (v), Chapter 5.

CHAPTER 6

1. 4. 7. 10. 13. 16. 19. 22. 25. 28. 31. 34. 37. 40. 43. 46. 49. 52. 54. 55.

4 4 -4 6 -2

-0.22 0.70 -2.22 0.566 16 0.008 8 0.977 0.125 0.0126 31.64 1 y=antilogb×xa (i) See Table 9.7 (ii) 31.6% 56. (i) 1.0 (iv) 10.1 57. (i) 13.0 (iv) 9.23

2. 5. 8. 11. 14. 17. 20. 23. 26. 29. 32. 35. 38. 41. 44. 47. 50. 53.

9 0 -3 4 0.60 0.18 2.30 -1.40 -6 81 0.0625 109 2 5 1999 3.0000004 ln y=ln a+b ln x

(ii) 4 (v) 2.22 (ii) 11.0 (v) 7.04

3. 6. 9. 12. 15. 18. 21. 24. 27. 30. 33. 36. 39. 42. 45. 48. 51.

2 5 -7 3 1.56 -0.18 4.48 2 1.098 128 2 1.0023 13 0.02 4002 5 or -2

(iii)

2.3

(iii)

11.48

93

Answers

Table 9.7 Answers to Question 55, Chapter 6.

Table 9.8 Answers to Question 58, Chapter 6.

58. 59. 60. 61. 62. 63.

See Table 9.8 10.6 Ethylamine gives pH 11.4 for a 0.01 mol/l solution. 4.34 Carboxyl group 99.999% ionised Amino group 99.5% ionised

64. ∆G°=1783 J/mol

CHAPTER 7

1. 1. 2. 2.25. 4. -1.467 5. y=x-1. 7. y=-0.25x+3.5 8. y=3x+2 10. y=-x-4 11. y=0.25x+4.25 13-16. See Graphs 9.1-9.4 17. See Graph 9.5. Gradient =E=18.2 l mmol-1 cm-1 =18.2×103 l mol-1 cm-1.

3. 6. 9. 12.

1.67 y=2.5x-6.5 y=2x+4 y=-0.6x

94

Easy Mathematics for Biologists

Graph 9.1

Graph 9.2

18. See Table 9.9 19. (i) 0.02 absorbance minute-1 (ii) 0.0267 µmol minute-1 (iii) 0.267 µmol minute-1 ml-1 saliva (iv) 0.1 µmol minute-1 ml-1 saliva

Answers

Table 9.9 Answers to Question 18, Chapter 7.

Graph 9.3

Graph 9.4

95

96

Easy Mathematics for Biologists

Graph 9.5

20. The graph is not a straight line, though it approximates to one when the substrate concentration is low. Therefore the rate of reaction is not proportional to the substrate concentration, though it approaches proportionality at low substrate concentrations.

CHAPTER 8

1–9. See Graphs 9.6–9.23 10. Plot mm contraction vs log [histamine]. The result is an S-shaped curve. Since the maximum contraction is 13.4 mm, the half-maximum effect is 6.7 mm, and at this point the log [histamine] is -5.59, so the [histamine] is 2.6 µmol/l. 11. Plot [bound hormone] vs [bound]/[free hormone]. Gradient=-1nmol/l=Kd The bound-intercept=10.5 fmol/ml=total number of receptors/ml so 10.5 fmol/ ml×6.02×1023=6.3×109 receptors/ml. 12. Plot ln dps vs time. Gradient=-0.433, y-intercept=19.3 therefore ln dps=-0.433(time)+19.3 (i) For t=half-life, ln dps=ln (2.4×108/2)=18.6 therefore 18.6=-0.433(t1/2)+19.3 t1/2=(19.3-18.6)/0.433=1.61 days

Answers

Graph 9.6

Graph 9.7

Graph 9.8

97

98

Graph 9.9

Graph 9.10

Graph 9.11

Easy Mathematics for Biologists

Answers

Graph 9.12

Graph 9.13

Graph 9.14

99

100

Graph 9.15

Graph 9.16

Graph 9.17

Easy Mathematics for Biologists

Answers

Graph 9.18

Graph 9.19

Graph 9.20

101

102

Graph 9.21

Graph 9.23

Graph 9.22

Easy Mathematics for Biologists

Answers

(ii) At t=2 weeks=14 days, ln dps =-0.433(14)+19.3=13.24 therefore activity=antiln 13.24=5.6×105 dps 13. Plot ln rate vs. 1/T. Gradient=-2.16×103=-Ea/R therefore Ea=2.16×103×8.3 J/mol=17.9 kJ/mol 14. Plot log10 MR vs log10 BM. Gradient=0.77, y-intercept=0.55 therefore log MR=0.77 log BM+0.55 therefore MR=3.55×BM0.77 15. See Table 9.10 Rate v=a[substrate]/(b+[substrate]) Plot vs ; vs [S]; and v vs From these graphs, a=approx. 0.73 µmol/min, b=approx 60 µmol/l.

Table 9.10 Answers for Question 14, Chapter 8.

103

INDEX

Absorbance, 44, 46, 57, 59, 70, 72 Absorption coefficient, 44, 46, 70, 72 Antilog, 50–53, 56 Atomic mass, 16 Avogadro’s number, 16 Axes, 59, 63, 66, 78

of straight lines, 62–66, 69, 70 Equilibrium constant, 45, 46, 58 Evaluating an expression, 43–45 Exponential equation, 75 Exponents, 2, 29–33, 49, 50 dividing and multiplying, 30

Base see also Logs and Exponents, mathematical, 29–32, 49, 50, 52, 56 chemical, 46, 54–56, 58 Base units, 3 Buffer solutions, 55

Factoring, 42 Factors, 5, 29, 30, 42 Formula weight, 16, 17, 26, 27, 37 Fractions, 4–6, 8, 10, 11 adding, 4 dividing, 6 equivalent, 4 multiplying, 5 subtracting, 5

Calibration curve, 69, 71 Common logs, 49 Concentrations, 2 percentage, 9, 10, 12, 15, 16 molar, 16–21, 23–27, 35, 37, 44–47 [H+], 53–55, 57–58 graphs of, 68–72, 73, 84–85 Conjugate base, 46, 54–56, 58 Converting units, 14–16 Coordinates, 59, 66, 70 Decimals, 2, 6, 7, 36 dividing and multiplying, 7 Denominator, 4–6, 8 Dependent variable, 66, 69, 84 Differential form, 80 Dilutions, 18–21, 23–25, 27, 38 fold, 19, 20, 23–25 serial, 19, 25 Dissociation constant, 46, 53 Dissociation, of a weak acid, 46, 53, 58 Equations equivalent, 39

Gradient, 61–66, 70, 73–75 Graphs, 59 non-linear, 73 of straight lines, 59–69 Henderson-Hasselbalch equation, 54, 55 Independent variable, 66, 69, 83 Integrated form, 80 Litre, equivalent of, 4, 35 Logarithmic equation, 75 Logarithms, 49–53, 56, 57 rules for manipulating, 51 Molarity, 16, 17 Mole, 16–18, 21 Molecular weight, 9, 16, 17

105

106

Index

Natural logs, 49, 52 Numerator, 4–6 Parentheses, multiplying or dividing numbers in, 40 Percentage applying to a number, 8 calculating a, 8 concentrations, see Concentrations converting from a, 8 increase or decrease, 8 converting to, 8 pH, 53–58 pKa, 53–58 Powers, 30–33 Prefixes, 33–35 Proportions, 2, 13–16, 41, 59 Quadratic equations, 42

Rates of change, 69, 79–81 of reaction, 2, 69, 71–75, 81, 85 Ratio, 3, 13, 60 Roots, 42, 51 Scientific notation, 32–33, 36 SI units, 3, 4, 16 Square, 3, 30 Standard free energy change, 58 Straight line, see Equations of straight lines Transformations, 75, 80–82 Transmittance, 57 Units, 2–4, 18, 29, 33–37, 44 Units, converting, 14–16